Displacement Continuity Research Articles

I. The effect of symmetric and spatially varying equilibria and flow on MHD wave modes: slab geometry

ABSTRACT Realistic theoretical models of magnetohydrodynamic wave propagation in the different solar magnetic configurations are required to explain observational results, allowing magnetoseismology to be conducted and provide more accurate information about local plasma properties. The numerical approach described in this paper allows a dispersion diagram to be obtained for any arbitrary symmetric magnetic slab model of solar atmospheric features. This proposed technique implements the shooting method to match necessary boundary conditions on continuity of displacement and total pressure of the waveguide. The algorithm also implements fundamental physical knowledge of the sausage and kink modes such that both can be investigated. The dispersion diagrams for a number of analytic cases that model magnetohydrodynamic waves in a magnetic slab were successfully reproduced. This work is then extended by considering density structuring modelled as a series of Gaussian profiles and a sinc(x) function. A further case study investigates properties of MHD wave modes in a coronal slab with a non-uniform background plasma flow, for which the governing equations are derived. It is found that the dispersive properties of slow body modes are more greatly altered than those of fast modes when any equilibrium inhomogeneity is increased, including background flow. The spatial structure of the eigenfunctions is also modified, introducing extra nodes and points of inflexion that may be of interest to observers.

A new hybrid homogenization theory for periodic composites with random fiber distributions

A new homogenization theory is constructed for unidirectional composites with periodic domains containing random fiber distributions. Periodic domains are partitioned into subdomains comprised of single fibers embedded in matrix phase. The subdomain displacement field solutions combine finite-volume and locally-exact elasticity approaches. Inclusions are regarded as meshfree components whose displacement fields are represented by discrete Fourier transforms that satisfy exactly Navier’s equations. In contrast, the matrix is discretized into subvolumes and handled within the finite-volume micromechanics framework. Novel use of traction and displacement continuity conditions at the inclusion/matrix interface seamlessly connect inclusion and matrix phases. Subdomains’ assembly enforces traction, displacement continuity and periodicity conditions in a surface-average sense. Quadratic convergence of stress fields with decreasing matrix discretization relative to exact elasticity solution is obtained, yielding accurate homogenized moduli with relatively coarse matrix meshes. Fourfold and greater reductions in execution times relative to the finite-volume micromechanics justify the new theory’s construction. The proposed theory facilitates accurate and efficient studies of the rarely reported effect of random fiber distributions on homogenized moduli and local stress fields as a function of inclusion content and number of microstructural realizations. We illustrate this by computing probability distributions of homogenized moduli for up to 60,000 microstructural realizations of multi-inclusion periodic domains that may be used in machine-learning algorithms.

Component-wise approach to reinforced concrete structures

With the rapid development of engineering constructions, especially transportation facilities, the structural models for the simulation of large-scale structures shall be eventually enhanced for predicting the complete three-dimensional stress and strain fields in reinforced concrete-made components. This paper proposes a component-wise approach for the modeling of reinforced concrete structures in which steel rebars and the concrete part are considered as two independent one-dimensional entities. Lagrange polynomials are used to express the cross-section deformations and different component-wise subdomains are joined by simply imposing displacement continuity at the chosen Lagrange points along the component boundary. The Finite Element (FE) method is applied to provide numerical solutions whereas Carrera Unified Formulation (CUF) is used to generate the related stiffness matrices in a compact and straightforward way. The classical case of homogenized beam solutions, as well as the one in which a virtual layer is associated with the steel zones, are implemented too. The three solutions are compared for a number of reinforced concrete beam problems, from single to double reinforced beam, including a T-shape cross-section. A final study considering transverse stiffeners (steel stirrups) is investigated. These stiffeners are modeled component-wise as well. Results clearly show the advantages and superiority of the component-wise FE-CUF based model to completely capture the three-dimensional strain and stress states, including shear ones, of reinforced concrete structures.

A semi-analytical method for vibro-acoustic analysis of submerged ring-stiffened cylindrical shells coupled with arbitrary inner structures

A semi-analytical method is developed for vibration and acoustic analysis of a submerged ring-stiffened cylindrical shell coupled with arbitrary inner structures. The whole structure is firstly disassembled into the stiffened cylindrical shell and inner structures, which are respectively analyzed through analytical substructure method and numerical approach. For the analytical substructure method, the cylindrical shell is decomposed into several shell segments according to positions of stiffeners and external forces, including exciting forces and coupling forces. These shell segments are described by Flügge shell theory and the ribs are governed by the annular plate equations. The solutions for cylindrical shell segments and rings are expanded as wave functions and Bessel functions. For the inner structures with arbitrary irregular shapes, finite element method (FEM) is utilized to establish their motion equations. At each coupling node, an artificial spring is adopted to connect the shell and inner structures, so rigid and elastic coupling conditions can be easily simulated by setting different stiffenesses to these springs. For external fluid field, an axisymmetric boundary element method (BEM) is employed to calculate the acoustic pressure. Combining the displacement continuity and force balance conditions at the joints of adjacent shell segments and introducing the coupling forces derived from inner structures and the fluid loading, the vibro-acoustic equations of the coupled system are obtained. To verify the validity of the present approach, vibro-acoustic responses of the semi-analytical method are compared with the ones calculated through FEM/BEM. Furthermore, effects of the coupling stiffness between the shell and inner structures and external fluid loading on vibro-acoustic responses are also evaluated.

Modeling of Thin Plate Flexural Vibrations by Partition of Unity Finite Element Method

This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM) for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope between elements is addressed by the use of cubic Hermite-type Partition of Unity (PU) functions. With appropriate PU functions, the PUFEM allows the incorporation of the special enrichment functions into the finite elements to better cope with plate oscillations in a broad frequency band. The enrichment strategies consist of the sum of a power series up to a given order and a combination of progressive flexural wave solutions with polynomials. The applicability and the effectiveness of the PUFEM plate elements are first verified via the structural frequency response. Investigation is then carried out to analyze the role of polynomial enrichment orders and enriched plane wave distributions for achieving good computational performance in terms of accuracy and data reduction. Numerical results show that the PUFEM with high-order polynomials and hybrid wave-polynomial combinations can provide highly accurate prediction results by using reduced degrees of freedom and improved rate of convergence, as compared with the classical FEM.

Isogeometric analysis of multi-patch solid-shells in large deformation

In the context of isogeometric analysis (IGA) of shell structures, the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state. However some basic questions remain unresolved when using solid-shell element, especially for large deformation cases with patch coupling, which is a common scene in real-life simulations. In this research, after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also presented to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The solid-shell elements are numerically studied by a linear elastic plate example, then we demonstrate the performance of the proposed formulation in large deformation, in terms of result verification, iteration history and continuity of displacement across the coupling interface. In the context of isogeometric analysis (IGA), after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also introduced to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The performance of the proposed formulation in large deformation is demonstrated by several examples.

Existence and uniqueness results on biphasic mixture model for an in-vivo tumor

In this article, we propose a mathematical model that describes hydrodynamics and deformation mechanics within a solid tumor which is embedded in or adjacent to a healthy (normal) tissue. The tumor and normal tissue regions are assumed to be deformable and the theory of mixtures is adapted to mass and momentum balance equations for fluid flow and tissue deformation mechanics in each region. The momentum balance equations are coupled via forces that interact between the phases (fluid and solid). Continuity of normal velocities, displacements, and normal stresses along with the Beaver–Joseph–Saffman condition are imposed at the interface between the tumor and tissue regions. The physiological transport parameters (such as hydraulic resistivity or permeability) are assumed to be heterogeneous and deformation dependent which makes the model nonlinear. We establish the existence of a weak solution using Galerkin and weak convergence methods. We show further that the solution is unique and depends continuously on the given data.

Donaldson Co Inc, USA

A semi-analytical method is developed for vibration and acoustic analysis of a submerged ring-stiffened cylindrical shell coupled with arbitrary inner structures. The whole structure is firstly disassembled into the stiffened cylindrical shell and inner structures, which are respectively analyzed through analytical substructure method and numerical approach. For the analytical substructure method, the cylindrical shell is decomposed into several shell segments according to positions of stiffeners and external forces, including exciting forces and coupling forces. These shell segments are described by Flügge shell theory and the ribs are governed by the annular plate equations. The solutions for cylindrical shell segments and rings are expanded as wave functions and Bessel functions. For the inner structures with arbitrary irregular shapes, finite element method (FEM) is utilized to establish their motion equations. At each coupling node, an artificial spring is adopted to connect the shell and inner structures, so rigid and elastic coupling conditions can be easily simulated by setting different stiffenesses to these springs. For external fluid field, an axisymmetric boundary element method (BEM) is employed to calculate the acoustic pressure. Combining the displacement continuity and force balance conditions at the joints of adjacent shell segments and introducing the coupling forces derived from inner structures and the fluid loading, the vibro-acoustic equations of the coupled system are obtained. To verify the validity of the present approach, vibro-acoustic responses of the semi-analytical method are compared with the ones calculated through FEM/BEM. Furthermore, effects of the coupling stiffness between the shell and inner structures and external fluid loading on vibro-acoustic responses are also evaluated.

Thermal shock response of porous functionally graded sandwich curved beam using a new layerwise theory

This paper, for the first time, presents the thermal shock response of porous functionally graded (FG) sandwich curved beams subjected to thermal shocks. The authors have used a higher-order layerwise beam theory developed for the analysis. This new layerwise theory has a higher-order displacement field for the core and a linear displacement field for top and bottom facesheets maintaining the continuity of displacement at the layer interface. The top and bottom layers of the functionally graded sandwich curved beam are made of pure ceramic and pure metal constituent, respectively, and the core is considered to be made of functionally graded material (FGM) having porosity. The top surface of the beam is subjected to thermal shocks the bottom surface is kept at a reference temperature or is thermally insulated. The solution for the nonlinear temperature profile is obtained using the Crank-Nicolson method. An eight-noded isoparametric element having seven degrees of freedom per node is used to develop the finite element formulation. The governing differential equation is obtained using the Hamilton’s principle, and the resulting transient problem is solved using the Newmark constant acceleration integration method. The effects of curvature, thickness, core to facesheet thickness ratio, intensity of the thermal shock, porosity coefficient, and boundary conditions are investigated on thermally induced vibration response of porous FG sandwich curved beams. The analysis reveals that the proposed finite element formulation is straight forward, accurate, and widely applicable for the analysis of porous functionally graded sandwich curved beams.

Modeling of cracks in two-dimensional elastic bodies by coupling the boundary element method with peridynamics

A multi-scale method based on a combination of the boundary element method (BEM) and peridynamics (PD) was developed to model crack propagation problems in two-dimensional (2D) elastic bodies. The special feature of this method is that it can take full advantage of both the BEM and PD to achieve a higher level of computational efficiency. Based on the scale of the structure and the crack location, the considered domain can be divided into non-cracked and cracked domains. The BEM is employed in the non-cracked domain, while the PD is applied in the cracked domain. This can reduce the dimension by one in the non-cracked domain for improving the modeling efficiency. A stiffness equation of the bond-based PD is established by using Taylor’s series expansion for the bond stretch and applied to simulate the cracked domain. The PD approach can automatically model the initiation and propagation of a crack. A coupling model using shared nodes is constructed by introducing the BEM nodes on the interface at the same location as the PD material points. With the continuity of displacements and equilibrium of tractions at the interface, a combined system of equations is obtained by merging the stiffness and force matrix from each domain. For test problems, the deformation and crack propagation in 2D elastic bodies subjected to quasi-static loads were analyzed. The numerical results clearly demonstrate the accuracy and efficiency of the proposed method for crack problems based on coupling the BEM and PD.

A practical method for seismic response analysis of nonlinear soil-structure interaction systems

Soil–structure interaction (SSI) plays an important role in the analysis of seismic structural responses. This study significantly extends an efficient linear SSI analysis method presented previously by the authors and co-workers to realistic nonlinear SSI systems, that is, systems with nonlinear soil, nonlinear structures, and flexible foundations (e.g. single- or multiple-pile foundations). The flexible foundations lying on half-space nonlinear soil are represented by frequency-dependent compliance functions that are fitted numerically instead of obtained by closed-form solution. These functions are then transferred to the time domain using the discrete-time recursive filtering method. A non-iterative algorithm is applied to guarantee the boundary conditions between soil and structure, that is, the displacement continuity and force equilibrium between them. The proposed method is implemented on an open-source FE software framework, called OpenSees. The accuracy and efficiency of the extended coupling method are investigated in detail through the seismic response analyses of typical soil–foundation–structure systems while considering the cases of linear or nonlinear soil, linear or nonlinear structures, and single- or multiple-pile foundations. Results show that the extended coupling method is significantly faster than the traditional FE method and provides acceptably accurate solutions for SSI systems with linear or low-to-moderate nonlinear soil. The paper provides a method for fast evaluation of nonlinear SSI effects in seismic structural response analysis.

An efficient eight‐node quadrilateral element for free vibration analysis of multilayer sandwich plates

AbstractThis article presents a free vibration analysis of laminated sandwich plates under various boundary conditions by using an efficient C0 eight‐node quadrilateral element. This new element is formulated based on the recently proposed layerwise model. The present model assumes an improved first‐order shear deformation theory for the face sheets while a higher‐order theory is assumed for the core maintaining an interlaminar displacement continuity. The advantage of this model relies on its number of variables is fixed, does not increase when increasing the number of lamina layers. This is a very important feature compared to the conventional layerwise models and facilitates significantly the engineering analysis. Indeed, the developed finite element is free of the shear locking phenomenon without requiring any shear correction factors. The governing equations of motion of the sandwich plate are derived via the classical Hamilton's principle. Several examples covering the various features such as the effect of modular ratio, aspect ratios, core‐to‐face thickness ratio, boundary conditions, skew angle, number of layers, geometry and ply orientations are solved for laminated composites and sandwich plates. The obtained results are compared with 3D, quasi‐3D, 2D analytical solutions, and those predicted by other advanced finite element models. The comparison studies indicate that the developed finite element model is of fast convergence to the reference and valid for both thick and thin laminated sandwich plates. Finally, it can be concluded that the present model is not only simple and accurate than the conventional ones, but also comparable with refined analytical solutions found in the literature.

On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates

This paper presents a novel C0 higher-order layerwise finite element model for static and free vibration analysis of functionally graded materials (FGM) sandwich plates. The proposed layerwise model, which is developed for multilayer composite plates, supposes higher-order displacement field for the core and first-order displacement field for the face sheets maintaining a continuity of displacement at layer. Unlike the conventional layerwise models, the present one has an important feature that the number of variables is fixed and does not increase when increasing the number of layers. Thus, based on the suggested model, a computationally efficient C0 eight-node quadrilateral element is developed. Indeed, the new element is free of shear locking phenomenon without requiring any shear correction factors. Three common types of FGM plates, namely, (i) isotropic FGM plates; (ii) sandwich plates with FGM face sheets and homogeneous core and (iii) sandwich plates with homogeneous face sheets and FGM core, are considered in the present work. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume power laws of the constituents. The equations of motion of the FGM sandwich plate are obtained via the classical Hamilton’s principle. Numerical results of present model are compared with 2D, quasi-3D, and 3D analytical solutions and other predicted by advanced finite element models reported in the literature. The results indicate that the developed finite element model is promising in terms of accuracy and fast rate of convergence for both thin and thick FGM sandwich plates. Finally, it can be concluded that the proposed model is accurate and efficient in predicting the bending and free vibration responses of FGM sandwich plates.

Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment

A mathematical model is developed, though this article, to investigate a vibrational behaviour of functionally graded (FG) cracked microbeam rested on elastic foundation and exposed to thermal and magnetic fields. The model includes a size scale effect and temperature dependent material properties, for the first time. The crack is modelled as a rotating spring, that is connecting the two parts of the microbeam at the crack’s position. The equation of motion of the FG microbeam is obtained by using the Euler-Bernoulli beam theory for kinematic assumption and nonlocal elasticity theory for size-dependency effects. The transverse Lorentz force induced from the magnetic field is derived using Maxwell's equations. By adding the effects of thermal loading and foundation parameters on the cracked micro beam, the motion equation of the entire system is obtained using the Hamilton’s principle and then solved with a Navier type solution method. Eight constraints equations are used to derived the frequency equation, which are boundary conditions at the end points and the displacement, slope, bending moment and transverse force continuity in the section where the crack is located. The resulting system of equations is solved sequentially, and natural frequencies and vibration modes of the cracked microbeam are obtained. The model is verified with previous published work. Numerical results are presented to illustrate influences of temperature, material composition, foundation parameters and magnetic field on the dynamics of the cracked FG microbeam.

Static and dynamic analyses of functionally graded sandwich skew shell panels

The present work is an attempt to develop a simple, accurate and widely applicable finite element formulation having a C0 continuity of transverse displacement at nodes, for static and dynamic analyses of functionally graded sandwich skew shell (FGSSS) panels. A layerwise displacement field based on first-order shear-deformation theory for each layer along with Sanders’ approximation has been adopted for the analysis. Compatibility conditions are imposed at the layer interfaces to satisfy displacement continuity. Two different configurations of FGSSS are taken up in the present investigation. In the first configuration, the top and bottom layers of the panel are made of functionally graded material (FGM) and the core is made of pure metal, whereas in the second configuration, the top and bottom layers are made of pure ceramic and pure metal, respectively and the core is considered to be made of FGM. The material properties of both configurations of FGM panels are estimated according to the rule of mixture. It is observed from the analysis that the present finite element formulation is simple, accurate and computationally efficient. Parameters like skew angle, span to thickness ratio, core to facesheet thickness ratio, volume fraction and boundary conditions have a significant effect on static and dynamic behavior of functionally graded sandwich skew shell panels.

Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

<abstract> <p>The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.</p> </abstract>

Buckling analysis of three-dimensional functionally graded sandwich plates using two-dimensional scaled boundary finite element method

For the stability analysis of functionally graded material (FGM) sandwich plates, a semi-analytical scaled boundary finite element method (SBFEM) is established within the frame work of layerwise theory in this paper. Two different configurations of FGM sandwich plates are studied, one of which consists of ceramic-rich core and FGM face sheets, and another involves metal-rich and ceramic-rich material face sheets as well as a FGM core, and the modulus of elasticity for each FGM layer is assumed to be changed continuously along thickness according to a power-law distribution. The layerwise theory performed in present work is based on the three-dimensional theory of elasticity for each individual layer satisfying the displacement continuity boundary conditions at layer interfaces, and the two-dimensional high order spectral element with three degrees of freedom per node is employed to discrete the reference surface of each individual layer. And then, based on the principle of virtual work, the SBFEM governing equation for each individual layer of FGM sandwich plates is derived, and a dual variable including the displacement and internal nodal force is used to reduce the governing equations to a system of first-order ordinary differential equation, which is solved analytically by a general approach. Finally, the critical buckling load can be obtained by solving the eigenvalue equation once the bending stiffness matrix and geometric stiffness matrix of FGM sandwich plates are determined. Fast rate of convergence of the proposed formulations is confirmed and comparison studies are presented to construct its high accuracy and excellent predictive capability. Furthermore, effects of gradient index, side-to-thickness ratio, layer configuration, and boundary conditions on dimensionless critical buckling loads of FGM sandwich plates are also studied. The proposed semi-analytical approach is simple in program implementation, accurate and computationally efficient.

Computational continua method and multilevel-multisite mesh refinement method for multiscale analysis of woven composites laminates

Three-scale computational continua (C2) method and multilevel-multisite (M2) mesh refinement method are developed for the multiscale analysis of woven composite laminates in this paper. Firstly, in order to carried out the full multiscale analysis of woven composite laminate, a three-scale C2 model is established. The cell with matrix and fiber, tow or yarn and laminated plate are considered as the microscopic, mesoscopic computational unit cells (CUC) and macroscopic problem, respectively. And then, a three-dimensional M2 method is developed to analyze the local displacements and stresses in the woven composite laminate. The refined meshes in the every scale unit cells are regarded as the super elements, and their contributions on the macroscopic stiffness are smeared into the total stiffness matrix of macroscopic problem. The displacement continuity and internal force equilibrium at the interface between two different scale CUCs are ensured by a continuity equation set, in which the requirement of meshing consistency between different CUCs is unnecessary, and the large scale effect can be considered between different CUCs. Several numerical examples are carried out to validate the proposed C2 and M2 methods by comparing with the direct numerical simulations (DNS) models.

Time Response Stress Analysis of Solid and Reinforced Thin-Walled Structures by Component-Wise Models

This paper deals with the evaluation of time response analyses of typical aerospace metallic structures. Attention is focussed on detailed stress state distributions over time by using the Carrera Unified Formulation (CUF) for modeling thin-walled reinforced shell structures. In detail, the already established component-wise (CW) approach is extended to dynamic time response by mode superposition and Newmark direct integration scheme. CW is a CUF-based modeling technique which allows to model multi-component structures by using the same refined finite element for each structural component, e.g. stringers, panels, ribs. Component coupling is realized by imposing displacement continuity without the need of mathematical artifices in the CW approach, so the stress state is consistent in the entire structural domain. The numerical results discussed include thin-walled open and closed section beams, wing boxes and a benchmark wing subjected to gust loading. They show that the proposed modeling technique is effective. In particular, as CW provides reach modal bases, mode superposition can be significantly efficient, even in the case of complex stress states.

Equivalent model of thin-walled orthotropic composite plates with straight segments using variational asymptotic method

Accurate and effective prediction of the static and dynamic behavior of thin-walled orthotropic composite plates (TW-OCP) is necessary in the preliminary design. Considering that the thickness of each segment of TW-OCP is relatively thin, the constitutive modeling of unit cell is carried out to obtain the constitutive relations by using variational asymptotic method, which can be used as effective properties of the two-dimensional equivalent plate model (2D-EPM). Furthermore, constraints are applied at the intersections of each segment to ensure the continuity of displacement and slope. Finally, the effects of different structural parameters on the equivalent stiffness and anisotropy are discussed, such as the layup configuration of the panel, the inclination angle and thickness of stiffeners, and the period length of unit cell. The results show that compared with the results from three-dimensional finite element model (3D-FEM), the 2D-EPM not only has high accuracy, but also greatly reduces the degree of freedom, which has a high computational efficiency. Geometric shape (mainly determined by the inclination angle of longitudinal stiffeners) is one of the main factors that determine the anisotropy and equivalent stiffness of TW-OCP. The established equivalent model provides a simple, quick, and comprehensive way for the design and optimization of TW-OCP.