Mixed Interpolation Of Tensorial Components Research Articles

Geometrically nonlinear analysis of plates and shells by a cell-based smoothed CS-MITC18+ flat shell element with drilling degrees of freedom

A development of a 3-node triangular flat shell element to analyze geometric nonlinearity of plate and shell structures is presented in this study. The proposed flat shell element has 6 degrees of freedom per node, including the drilling rotational ones, by using the Allman-type approximations for the in-plane displacements and the bending displacement approximations enriched by a cubic shape function at the bubble node located at the element's centroid. The geometrically nonlinear behaviors are described by the von Karman's large deflection assumption. The membrane and bending strains of the presented flat shell elements are averaged over sub-triangular elements based on the cell-based smoothed (CS) technique. To attenuate the shear locking phenomenon, the transverse shear strains are re-interpolated following the mixed interpolation of tensorial components (MITC) technique designed for the 3-node triangular degenerated shell elements enriched by a cubic bubble function (MITC3+). Combined with the Newton-Raphson iterations and load increments predicted by the arc-length method, the suggested 3-node triangular flat shell elements, namely the CS-MITC18+ element, can analyze moderately geometrical nonlinearity, including the snap-thought and snap-back behaviors, of various plates and shells with the different shapes, thickness, and boundaries. The numerical investigations show that the load-displacement curves provided by the CS-MITC18+ flat shell elements well agree with other references.

Nonlinear static analysis of functionally graded porous sandwich plates resting on Kerr foundation

The main purpose of this work is to explore the nonlinear static bending analysis of functionally graded porous sandwich plates with a ceramic core (FSCC) subjected to a transverse uniform load resting on Kerr foundation (KF) using the mixed interpolation of tensorial components for the four-node rectangular element (called MITC4). Porosity appearing in two skin layers is assumed according to the uneven porosity distribution with a porosity factor The advantage of this element is fast convergence, high accuracy and cancels the shear-locking phenomenon. The present study considers the geometrical nonlinearity resulting from mid-plane stretching based on the von Kármán geometrical nonlinearity. The first-order shear deformation theory (FSDT) is employed to approximate the displacement field of the plate due to its simplicity and efficiency. The current formulation’s authenticity is evaluated by comparing it with previously published works. For the first time, the nonlinear bending of FSCC subjected to a transverse uniform load resting on the Kerr foundation is investigated. Furthermore, the influence of input parameters on the nonlinear bending behavior of FSCC is fully provided.

A triangular inverse element coupling mixed interpolation of tensorial components technique for shape sensing of plate structure

This novel inverse element, named as iMITC3+, is proposed by coupling the inverse finite element method (iFEM) and mixed interpolation of tensorial components (MITC) technology for real-time reconstructing the structural deformation based on discrete strain measurement. This reconstructed method is established through combining the kinematics of enhanced 3-node triangular MITC element and least-squares variational function. Furthermore, the approach containing the cubic bubble function inside the element is less sensitive to deformation under bending-dominated and thus alleviates the shear locking for shape sensing of thin plate structure. In addition, the method is suitable for geometrically arbitrarily plate structures, due to that the rotation degree of freedom of the element is represented by the nodal director vector. The high accuracy and practicability of iMITC3+ has been verified by numerical and experimental results through analyses of plate model subjected to different loading.

A fully coupled 3D elastohydrodynamic model built with MITC element for static performance analysis of gas foil bearings

PurposeThis paper aims to present a 3D static performance analysis model for the gas foil journal bearing to provide better understanding of the gas foil journal bearing and extend the development of the calculation about the static performance.Design/methodology/approachThe foil bearing can be seen as a shell structure, and the mixed interpolation of tensorial components (MITC) element was used to build the shell model. The augmented Lagrange method was used to calculate the contact involving friction between foils and between the foil and the bearing sleeve. A displacement-controlled load scheme was used to calculate the deformation of the foils. A mapping operator was used to map the film pressure from the gas to the surface of the top foil.FindingsThis method provides high precision of calculation in the prediction of the static performance. The calculation results were compared with the experimental data, and they show good agreement. Meanwhile, the model can be applied in the prediction of the bearing performance in a broad range of working conditions.Originality/valueThis method extends the calculation of the gas foil journal bearing to a 3D scale and shows good agreement with the experimental data. Meanwhile, the present model has a good adaptability on the revolution speed and can be applied to the predictions in varied working conditions.

Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements

The main goal of this article further extends the ES-MITC3 element based on first-order shear deformation theory (FSDT) for the free vibration analysis of functionally graded porous (FGP) plates resting on the partially supported elastic foundation (PSEF). The ES-MITC3 element is the union between the mixed interpolation of tensorial components (MITC) technique and an edge-based smoothed finite element method (ES-FEM). The elastic foundation (EF) is a Winkler-Pasternak’s model with two parameters consisting of Winkler-stiffness ( k 1 ) and Pasternak-stiffness ( k 2 ). The porosity distribution in plates is assumed to vary according to the even distribution (case 1) and uneven distribution (case 2) rules. The governing equation is derived from Hamilton's principle. Numerical examples are performed to verify the accuracy and reliability of the present method. In addition, the effects of types of PSEF, material properties and geometry characteristics on the free vibration behavior of FGP plates are fully investigated.

A higher-order quadrilateral shell finite element for geometrically nonlinear analysis

In this paper, we present a geometrically nonlinear formulation for a nine-node shell finite element and demonstrate its performance. The total Lagrangian formulation is utilized to allow for large displacements and large rotations. The mixed interpolation of tensorial components (MITC) technique is applied to effectively reduce the membrane and shear locking phenomena without refining mesh, introducing additional nodes, and using reduced integrations. Many-core accelerators with GPU-compatible libraries are used to achieve the highest speed-up for the solution process. The performance of the present nine-node shell element is compared with those of the four-node and six-node shell elements in several benchmark problems, including Cook's skew beam, cantilever plate, pinched cylindrical shell, slit annular plate, and hemisphere shell. The present element achieves excellent performance even when the coarse mesh is used.

Isogeometric MITC shell

This paper proposes an isogeometric formulation of the Reissner–Mindlin shell, using the Mixed Interpolation of Tensorial Components (MITC) technique to alleviate shear locking and membrane locking. Instead of over each Non-Uniform Rational B-Spline (NURBS) element, kinematics of the shell is directly formulated on the entire NURBS patch, with its displacements decoupled to the motions of control points and of director vectors tied to the control points. Since control points are in general not located on the surface, those control-point director vectors are interpolated from normal vectors at a set of pre-defined integration points on the patch. The assumed in-plane membrane strain field and the assumed transverse shear strain field are built as linear combinations of NURBS basis functions with degrees lower than those employed by the displacement interpolation, and tied to the original covariant strain fields at a set of well-selected tying points. Integration points for the definition of control-point director vectors and tying points of the assumed covariant strain fields are provided by the optimal and reduced macro-element quadrature rules, respectively. In this way, the classical MITC technique is fully incorporated into the isogeometric framework with arbitrary polynomial orders or patch configurations. Thanks to the high smoothness of NURBS basis functions, the geometry-preserving nature of isogeometric transformation, and the locking-resistive capability of MITC technique, the isogeometric MITC shell formulation shows superior convergence behavior, minimal geometrical error, and good robustness against patch distortion. In particular, its convergence property is insensitive to NURBS polynomial order or control point density. These advantages are demonstrated through a number of well-established benchmark problems, validating the proposed isogeometric MITC formulation being an accurate and efficient solution for linear analysis of shell structures.

An efficient curved beam element for thermo-mechanical nonlinear analysis of functionally graded porous beams

This research is aimed to develop an efficient and high-performance four-node iso-parametric beam element, which is composed of Functionally Graded Material (FGM). In addition, different patterns of material distribution will be considered through the height of element. On the other hand, beam’s imperfection, presented here with porosity, is taken into account by using the rule of mixture. In order to alleviate the shear locking, Mixed Interpolation of Tensorial Components (MITC) is utilized by using tying points. Strain interpolation at some tying points reduces the order of strain functions. Therefore, three Gauss points can be employed for numerical integration instead of four Gauss points. Furthermore, the geometrically nonlinear effects are incorporated by using Green-Lagrange strains. Since the material properties are considered to be thermal-independent, they remain constant during the analysis. Finally, some benchmark problems are solved to illustrate the correctness of formulation and accuracy of the proposed element. Several parameters, including porosity percentage, FGM patterns and corresponding power indices, are investigated in the other examples. It is observed that the proposed element is more accurate for linear and nonlinear analyses of the thin beam with large deformations and rotations, even by using fewer numbers of elements compared to other developed elements. On the other hand, both axial and transverse displacements decrease when the value of the exponent of sigmoid pattern increases. On the contrary, the exponent of power pattern has a different effect on the axial displacement.

A Finite Element Formulation and Nonlocal Theory for the Static and Free Vibration Analysis of the Sandwich Functionally Graded Nanoplates Resting on Elastic Foundation

This article presents a finite element method (FEM) integrated with the nonlocal theory for analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation (EF). Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). In this study, the formulation of the four-node quadrilateral element based on the mixed interpolation of tensorial components (MITC4) is used to avoid “the shear-locking” problem. On the basis of Hamilton’s principle and the nonlocal theory, the governing equations for the sandwich FG nanoplates are derived. The results of the proposed model are compared with published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and material properties on the static and free vibration behaviors of nanoplates are investigated in detail.

Formulation of 3D finite elements using curvilinear coordinates

The objective of this paper is to gain insight into the implementation of 3D finite elements in curvilinear coordinates using the fundamental equations of 3D elasticity and the Principle of Virtual Displacements. After reviewing the mathematical model of the geometry, we present the formulation of hexahedral finite elements that represent a generalization of classical shell finite elements, in which also the thickness direction can be curvilinear. These finite elements are equivalent to the widely-used continuum mechanics based solid finite elements, but the adoption of local curvilinear reference system will allow in future works to apply mixed methods, such as the Mixed Interpolation of Tensorial Components (MITC), to contrast the locking phenomenon in the three spatial directions. For the assessment of the present elements, we consider different examples of curved geometries: cylindrical, conical, toroidal and spherical. A free-vibration analysis of curved 3D components is performed and the results, in terms of natural frequencies, are compared with the convergence solutions computed with the commercial software MSC Patran/Nastran.

Multiphase material topology optimization of Mindlin-Reissner plate with nonlinear variable thickness and Winkler foundation

In typical, structural topology optimization plays a significant role to both increase stiffness and save mass of structures in the resulting design. This study contributes to a new numerical approach of topologically optimal design of Mindlin-Reissner plates considering Winkler foundation and mathematical formulations of multi-directional variable thickness of the plate by using multi-materials. While achieving optimal multi-material topologies of the plate with multi-directional variable thickness, the weight information of structures in terms of effective utilization of the material at the appropriate thickness location may be provided for engineers and designers of structures. Besides, numerical techniques of the well-established mixed interpolation of tensorial components 4 element (MITC4) is utilized to overcome a well-known shear locking problem occurring to thin plate models. The well-founded mathematical formulation of topology optimization problem with variable thickness Mindlin-Reissner plate structures by using multiple materials is derived in detail as one of main achievements of this article. Numerical examples verify that variable thickness Mindlin-Reissner plates on Winkler foundation have a significant effect on topologically optimal multi-material design results.

Assessment of MITC plate elements based on CUF with respect to distorted meshes

This paper discusses the robustness of plate elements based on Mixed Interpolation of Tensorial Components (MITC) technique and the variable kinematics approach of Carrera Unified Formulation (CUF) with respect to the problem of distorted meshes. MITC was originally proposed for Reissner-Mindlin type plates to develop shear locking free plate elements. In the present framework, refined plate elements are obtained by referring to high-order Equivalent Single Layer as well as Layer-Wise models expressed in CUF for the analysis of multilayered anisotropic structures. Four-node and nine-node elements are considered and some applications are developed for both isotropic and multilayered composite plates. Results related to the MITC approach are compared to the isoparametric elements, including selectively reduced quadrature schemes, for both, the static and free-vibration analysis. They show that CUF-MITC elements maintain their effectiveness also in the case of distorted meshes, for all the materials studied and kinematic models considered: the obtained elements are robust as free from shear locking and spurious zero-energy modes.

Hybrid‐Trefftz stress elements for plate bending

SummaryThe hybrid‐Trefftz stress element is used to emulate conventional finite elements for analysis of Kirchhoff and Mindlin‐Reissner plate bending problems. The element is hybrid because it is based on the independent approximation of the stress‐resultant and boundary displacement fields. The Trefftz variant is consequent on the use of the formal solutions of the governing Lagrange equation to approximate the stress‐resultant field. In order to emulate conventional elements, nodal functions are used to approximate the displacements on the boundary of the element. Duality is used to set up the element solving system. The associated variational statements and conditions for existence and uniqueness of solutions are recovered. Triangular and quadrilateral elements are tested and characterized in terms of convergence, sensitivity to shear‐locking, and shape distortion. Their relative performance is assessed using assumed strain Mixed Interpolation of Tensorial Components (MITC) elements and recently proposed Trefftz‐based elements. This relative assessment is extended to a hypersingular problem to illustrate the effect of enriching the domain and boundary approximation bases.

A Mixed Stress/Displacement Approach Model of Homogeneous Shells for Elastodynamic Problems

This paper presents the development of a model of homogeneous, moderately thick shells for elastodynamic problems. The model is obtained by adapting and modifying SAM-H model (stress approach model of homogeneous shells) developed by Domínguez Alvarado and Díaz in (2018) for static problems. In the dynamic version of SAM-H presented herein, displacements and stresses are approximated by polynomials of the out-of-plane coordinate. The stress approximation coincides with the static version of SAM-H when dynamic effects are neglected. The generalized forces and displacements appearing in the approximations are the same as those involved in a classical, moderately thick shell model (CS model) but the stress approximation adopted herein is more complex: the 3D motion equations and the stress boundary conditions at the faces of the shell are verified. The generalized motion and constitutive equations of dynamic SAM-H model are obtained by applying a variant of Euler–Lagrange equation which includes pertinently Hellinger–Reissner functional. In the constitutive equations, Poisson’s effect of out-of-plane normal stresses on in-plane strains is not ignored; this is one important feature of SAM-H. To test the accuracy of dynamic SAM-H model, the following structures were considered: a hollow sphere and a catenoid. In each case, eigenfrequencies are first calculated and then a frequency analysis is performed applying a harmonic load. The results are compared to those of a CS model, MITC6 (mixed interpolation of tensorial components with 6 nodes per element) shell element calculations, and solid finite element computations. In the two problems, CS, MITC6, and dynamic SAM-H models yield accurate eigenfrequencies and eigenmodes. Nevertheless, the frequency analysis performed in each case showed that dynamic SAM-H provides much more accurate amplitudes of stresses and displacements than the CS model and the MITC6 shell finite element technique.

New higher-order triangular shell finite elementsbased on the partition of unity

Finite elements based on the partition of unity (PU) approximation have powerful capabilities for p-adaptivity and solutions with high smoothness without remeshing of the domain. Recently, the PU approximation was successfully applied to the three-node shell finite element, properly eliminating transverse shear locking and showing excellent convergence properties and solution accuracy. However, the enrichment with the PU approximation results in a significant increase in the number of degrees of freedom; therefore, it requires greater computational cost, thus making it less suitable for practical engineering. To circumvent this disadvantage, we propose a new strategy to decrease the total number of degrees of freedom in the existing PU-based shell element, without loss of optimal convergence and accuracy. To alleviate the locking phenomenon, we use the method of mixed interpolation of tensorial components and perform convergence studies to show the accuracy and capability of the proposed shell element. The excellent performances of the new shell elements are illustrated in three benchmark problems.

A novel hybrid shell element formulation (QUAD+ and TRIA+): A benchmarking and comparative study

This paper introduces a novel hybrid finite element (FE) formulation of shell element to enable assembly process simulation of compliant sheet-metal parts with higher efficiency and flexibility. Efficiency was achieved by developing both new hybrid quadrilateral and triangular elements. Quadrilateral element (QUAD+) was formulated by combining area geometric quadrilateral 6 (AGQ6) nodes and mixed interpolated tensorial components (MITC) to model membrane and bending/shear component respectively. Triangular element (TRIA+) was formulated by merging assumed natural deviatoric strain (ANDES) for membrane and MITC for bending/shear component. Flexibility was addressed by developing an open-source C++ code, enhanced by the OpenMP interface for multiprocessing programming. Tests and benchmarks were compiled and executed within Matlab using the MEX API interface. Extensive benchmark studies were accomplished to evaluate the performance of the proposed hybrid formulation and the shell formulations used in three FEM packages - ABAQUS, ANSYS and COMSOL- under static linear elastic condition with small strain assumption. It was observed that the proposed QUAD+ and TRIA+ elements performed better amongst the FE packages, especially when there was in-plane mesh distortion, with errors below 3%. It was also identified that the best efficiency is obtained by adopting dominant QUAD+ elements compared to the TRIA+ when working on complex geometries. This paper also contributes to present a wide set of benchmark studies required to verify new release of FE packages using shell element or evaluate the performance of new shell formulations.

Static analysis of Reissner-Mindlin plates using ES+NS-MITC3 elements

In this research, the smoothed finite element methods (S-FEM) based on the edge-based (ES) and node-based (NS) approaches are combined to develop for the 3-node triangular plate element which uses the mixed interpolation of tensorial components (MITC3) technique to remove the shear-locking phenomenon. This approach is based on the βFEM in which the parameter β is used to tune the contribution ratio of the edge-based and node-based smoothed domains. The strain fields of the proposed ES+NS-MITC3 element are smoothed on a part of the edge-based domains and the other on the node-based domains which are respectively defined by elements sharing common edges and common nodes. The ES+NS-MITC3 element passes the patch test and is employed to statically analyze some benchmark Reissner-Mindlin plates, including square and rhombus ones. Numerical results show that, in both thin and thick plates the ES+NS-MITC3 element can give results better than similar elements using the ES-FEM or NS-FEM only.

An alternative approach for modal analysis of stiffened thin-walled structures with advanced plate elements

Nowadays, many vehicles and buildings are built using the structural concept of stiffened structure, in order to achieve greater strength with relatively less material. In the present work, a plate finite element with advanced higher-order kinematic field is proposed for the free-vibration analysis of stiffened thin-walled plate structures. A novel approach is introduced to simplify the modelization and assembling phase for the analysis of stiffened structures. The mechanical continuity between skin and stiffener deformed geometry is automatically verified with the present approach. On the basis of the Principle of Virtual Displacements, Equivalent-Single-Layer and Layer-Wise models related to linear up to fourth order variations of the unknown variables in the thickness direction are treated. The Mixed Interpolated Tensorial Components (MITC) method, in conjunction with the Finite Element Method (FEM), is employed to contrast the shear locking effect. Various stiffener shapes are considered for the assessment of the approach: T shape, double-T shape and Z shape. Several numerical investigations are carried out to validate and demonstrate the accuracy and efficiency of the present plate finite element for the modal analysis of stiffened structures, comparing the present results with those from classical finite element formulations based on solid elements available in commerical software.

Linear and geometrically nonlinear analysis of plane structures by using a new locking free triangular element

This research is dedicated to propose a high-performance locking-free triangular plane element, which is based on the novel mixed finite element formulation. The new element has six nodes with two transitional degrees of freedom at each node. To improve the displacement and stress responses, a proper mixed interpolation of the strain fields is propounded. Moreover, Mixed Interpolation of Tensorial Components (MITC) has been employed to alleviate in-plane shear locking. The advantage of the employed method is that no additional degrees of freedom are introduced, and spurious instabilities have not been seen. The tensorial forms of the equations are employed in this paper to summarize the formulations. The element passes the numerical tests, which are required to show the accuracy, capability and convergence rate in the structural analysis. Both linear and geometrically nonlinear problems can be solved by the presented formulations. To incorporate the large deformations, Total-Lagrangian formulation will be utilized. Due to completely trace the equilibrium paths of plane problems, especially those with snap-through behavior, a Generalized Displacement Control Method (GDCM) is used as a nonlinear solver. Outcomes of several benchmark problems and some new plane structures, which are analyzed separately, illustrate the high accuracy and advantages of the proposed element, especially in curved structures and nonlinear problems.

Sound Transmission Analysis of Viscoelastic Composite Multilayered Shells Structures

In the development of aircraft comfort, one of the main issues is the sound transmission analysis to estimate the insulation capability of aeronautical panels. In this work, a higher-order shell finite element is proposed for the passive noise insulation analysis of composite laminated structures embedding viscoelastic layers. Starting from the Principle of Virtual Displacements, the present Finite Elements are obtained by making use of higher-order Layer-Wise theories, employing the Mixed Interpolated Tensorial Components (MITC) method to avoid the shear locking effect and taking into account the frequency dependence of the viscoelastic material through the use of a fractional derivative model. The Rayleigh integral method is considered for the evaluation of the acoustic insulation of the panels. Numerical studies are carried out to demonstrate that the present shell finite element is an efficient and accurate tool for the sound transmission analysis. Different lamination sequences, different boundary conditions and various radius to thickness ratios are taken into account.