Mindlin Plate Element Research Articles

Richardson extrapolation based integration scheme using FSDT over Quad elements: Application to thick FGM plates

In this work, our new quadrature schemes based on the Richardson extrapolation (RE) and centroid or midpoint rule are further applied to the quadrilateral Mindlin plate element for static and free vibration analysis. In the process of generating element matrices, to apply RE and midpoint quadrature rule, there are two approximations are particularly considered. As the first approximation, element matrices are calculated at centroid of whole plate element. For stabilizing function or second approximation, each quadrilateral element is divided into four sub-quadrilaterals, either centroid of the each sub-quadrilateral plate for Element midpoint method (EM-plate method) or midpoint of the element edges are used for Element edge method (EE-plate method) to compute the all the element matrices. Then, both the approximations are added with the RE based weighting functions. Generally, midpoint based quadrature avoids the shear-locking phenomenon in the plate elements and the RE enhances accuracy and rate of convergence of the final solution without hourglass or zero-energy issues. The numerical examples validated that the RE based EM-plate method and EE-plate method are free of shear-locking, ability to pass the patch test, better rate of convergence with less number of sampling points. Also, the new schemes confirms the three important properties such as (i) can produce high accurate solutions with better rate of convergence for problems for irregular geometries in the static analysis; (ii) can produce high accurate solutions without hourglass issues in free vibration analysis; and (iii) can generate the accurate values of high frequencies of plate elements over irregular meshes.

Physics-Based Self-Learning Recurrent Neural Network enhanced time integration scheme for computing viscoplastic structural finite element response

In the current study, we present an application to the class of deep learning known as the Physics Informed Neural Networks (PINNs), more specifically we develop a new implicit material integration scheme based on Recurrent Neural Network (RNN). In this study, we refer to it as the Recurrent Enhanced Implicit Integration Scheme (REIIS). We illustrate how to incorporate the constitutive relations into REIIS and explore its application to Lemaitre–Chaboche viscoplastic model. Whilst common applications in this class of deep learning utilize feed-forward neural networks (FFNN) for training, we propose a combination of the Long Term Short Memory (LSTM) network with FFNNs for an accurate representation of the state variables. Furthermore, with the present strategy, we combine the data-driven approach of replacing the material behavior with NN along with its ability to learn continuously through physical laws. To validate the approach, we test the results of the framework with the numerical solution obtained with Finite Element Method (FEM) for Reissner–Mindlin plate elements at different strain rates. We further observe that obeying physical constraints leads to improved robustness in the learning of REIIS and broadens the scope of integrating the NNs in FEM.

Study on the Vector Quadrilateral Isoparametric Shell Element with Shear Deformation

Based on the linear superposition of the quadrilateral isoparametric membrane element and quadrilateral Mindlin plate element, basic vector formulas for the quadrilateral isoparametric shell element with the shear deformation is derived. Two different coordinate modes for updating the particle displacement and the element node internal force are put forward. The uniform numerical solution for an irregular quadrilateral element is realized by the isoparametric integral scheme along the element plane. Then, the bilinear elastoplastic constitutive of material is introduced into the vector finite shell element, and the nonlinear effect of the material is realized based on the integral method along the thickness. On this basis, the nonlinear calculation and analysis program of the quadrilateral isoparametric shell element is developed. Numerical example results show that the static and dynamic analysis, large deformation and large rotation analysis, and buckling analysis can all be well performed for the shell structures by the developed program, which verifies the validity of theoretical derivation and computer program.

Wave energy extraction and hydroelastic response reduction of modular floating breakwaters as array wave energy converters integrated into a very large floating structure

Combing floating breakwaters with wave energy converters (WECs) and integrating them into very large floating structure (VLFS) can provide a viable option to explore economically offshore wave energy resources and simultaneously to protect marine structures. In this paper, the time-domain numerical model is developed based on the modal expansion theory with nonlinear consideration to optimize the design and layout of an integrated system of modular WEC-type floating breakwaters and a pontoon-type VLFS, with emphasis on the effects of the WEC geometric size and shape, the WEC-VLFS gap distance and the wave nonlinearity. A hybrid finite element (FE)-boundary element (BE) method is presented to simulate the structures as Mindlin plate elements and the water waves as fully nonlinear potential flow boundaries, respectively. Breakwaters as WECs with deeper draft and larger length are found to more fully interact in phase with long-period waves, and receive more wave energy extraction and larger hydroelastic response reduction. The addition of breakwaters has a favorable effect on the wave energy extraction, but a destructive effect on the hydroelastic reduction. Importantly, wave resonance induced by the multi-modal scattering waves in the WEC-VLFS gap leads to multiple peaks of the power capture efficiency. Compared to the symmetric-shape WECs, the asymmetric-shape WECs strengthen the gap resonant effect, which improves both the wave energy extraction and hydroelastic reduction for a broader frequency bandwidth. The findings of this study indicate the synergistic benefits of wave energy exploitation and transmitted wave attenuation at the fore-end of VLFSs.

Temporal and spatial-domain DFT-based spectral element model for the dynamic analysis of a rectangular Mindlin plate

A temporal and spatial-domain spectral element model for the dynamics of a rectangular Mindlin plate based on the discrete Fourier transform was developed as an extension of the authors’ previous work. The following procedure was implemented to derive the spectral element model. First, the general solutions of a rectangular Mindlin plate element were derived in spectral forms in both the temporal and spatial domains. Second, the boundary functions that describe the boundary conditions were then expressed in spectral forms. Third, using the orthogonality properties of the trigonometric functions, the relationships between the spectral components of the general solutions and boundary functions were derived in the square-matrix-equation form. Finally, the spectral matrix was formulated for the entire plate element using the force-displacement relationships. The proposed spectral element method could provide accurate dynamic responses and waves in a plate structure using the fast Fourier transform algorithm owing to the spectral form representations of the general solutions and boundary functions in both the temporal and spatial domains. The high accuracy and computational efficiency of the proposed SEM were investigated by comparing the results with those of the exact theory, modal analysis method, Rayleigh-Ritz method, dynamic stiffness method, and commercial finite element analysis software ANSYS.

Optimization and analysis of frequencies of multi-scale graphene/fibre reinforced nanocomposite laminates with non-uniform distributions of reinforcements

Optimal design and analysis of three-phase graphene/fibre reinforced laminated nanocomposite plates with respect to maximizing the fundamental frequency is the subject of the present study. Optimal design solutions are given for four different sets of design parameters. First design problem determines the optimal graphene contents of individual layers, the second one both graphene and fibre contents, the third optimizes the graphene and fibre contents as well as the layer thicknesses of individual layers, and the fourth problem optimizes the graphene and fibre contents, layer thicknesses and fibre orientations. Purpose of this approach is to assess and compare different levels of optimization by means of a design efficiency index and as such to determine the effectiveness of different design parameters in maximizing the fundamental frequency. Optimization is implemented using a Sequential Quadratic Programming algorithm and the mechanical properties of graphene/fibre nanocomposite are determined via micromechanical relations. Vibration analysis is conducted by the finite element method using four-noded Mindlin plate elements. Results are obtained for simply supported (SSSS), clamped (CCCC) and simply supported-clamped boundary conditions for opposite edges (SCSC). It is observed that non-uniform distributions of graphene and fibre as well as fibre orientations are quite effective in improving the design efficiency.

Adjusted approximation spaces for the treatment of transverse shear locking in isogeometric Reissner–Mindlin shell analysis

Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided.

Generalized modal element method: part II—application to eight-node asymmetric and symmetric solid-shell elements in linear analysis

In this paper, two eight-node asymmetric solid-shell elements are firstly presented to illustrate the use of traditional finite element technique in GMEM for constructing new finite element formulations. In these two elements, the analytical method is utilized to derive the displacement functions of the basic in-plane modes, which makes the resulted elements free of mesh distortions for these modes. For out-of-plane modes, the Mindlin plate elements CQUAD4 and QTS4 are integrated to calculate the corresponding modal displacement vectors and modal force vectors of solid-shell elements. As a result, the displacement functions of the plate-bending element range up to three orders of magnitude. On the other hand, since the asymmetric elements cannot be applied to frequency analysis, two symmetric solid-shell elements are proposed by using a modal transformation method, in which the in-plane modes are derived from the previous proposed symmetric solid element S-MEM8S and the out-of-plane modes are constructed by plate elements CQUAD4 and QTS4, respectively. Various benchmarks including linear static and frequency analysis are presented to demonstrate the accuracy and efficiency of the presented elements.

Construction and Application of Multivariable Wavelet Finite Element for Flat Shell Analysis

Based on B-spline wavelet on the interval (BSWI) and the multivariable generalized variational principle, the multivariable wavelet finite element for flat shell is constructed by combining the elastic plate element and the Mindlin plate element together. First, the elastic plate element formulation is derived from the generalized potential energy function. Due to its excellent numerical approximation property, BSWI is used as the interpolation function to separate the solving field variables. Second, the multivariable wavelet Mindlin plate element is deduced and constructed according to the multivariable generalized variational principle and BSWI. Third, by following the displacement compatibility requirement and the coordinate transformation method, the multivariable wavelet finite element for flat shell is constructed. The novel advantage of the constructed element is that the solving precision and efficiency can be improved because the generalized displacement field variables and stress field variables are interpolated and solved independently. Finally, several numerical examples including bending and vibration analyses are given to verify the constructed element and method.

Static analysis of corrugated panels using homogenization models and a cell-based smoothed mindlin plate element (CS-MIN3)

Homogenization is a promising approach to capture the behavior of complex structures like corrugated panels. It enables us to replace high-cost shell models with stiffness-equivalent orthotropic plate alternatives. Many homogenization models for corrugated panels of different shapes have been proposed. However, there is a lack of investigations for verifying their accuracy and reliability. In addition, in the recent trend of development of smoothed finite element methods, the cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) has been proposed and successfully applied to many analyses of plate and shell structures. Thus, this paper further extends the CS-MIN3 by integrating itself with homogenization models to give homogenization methods. In these methods, the equivalent extensional, bending, and transverse shear stiffness components which constitute the equivalent orthotropic plate models are represented in explicit analytical expressions. Using the results of ANSYS and ABAQUS shell simulations as references, some numerical examples are conducted to verify the accuracy and reliability of the homogenization methods for static analyses of trapezoidally and sinusoidally corrugated panels.

Free vibration analysis of corrugated panels using homogenization methods and a cell-based smoothed Mindlin plate element (CS-MIN3)

The behavior analysis of complex structures like corrugated panels usually requires high-cost shell modeling which is an obstacle in subsequent studies such as parametric study and optimization. To overcome the difficulty, many homogenization models in which equivalent orthotropic plates replaces the original shells have been proposed. However, there is a lack of investigations for verifying their accuracy and reliability. In this paper, the cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) is integrated with typical homogenization models to give some homogenization methods for free vibration analyses of trapezoidally and sinusoidally corrugated panels. Using the results of ANSYS and ABAQUS shell simulations as references, the accuracy and reliability of the homogenization methods are illustrated through several numerical examples. The influences of geometric parameters such as the number of corrugation units, corrugation amplitude, trough angle, and panel thickness are studied and some important remarks about the performances of these methods are derived.

The quadrilateral Mindlin plate elements using the spline interpolation bases

By the Mindlin/Reissner plate theory, the displacement ω and rotations θ x , θ y are interpolated by independent functions, for which only C 0 continuity condition is required. The difficulty for constructing interpolation bases on the quadrilateral element without isoparametric transformation can be overcome by using the spline method. In this paper, two sets of spline interpolation bases are adopted to construct two quadrilateral spline Mindlin plate elements (QSMP1 and QSMP2) with 12 degrees of freedom. The spline elements can be applied for both thick and thin plates, and can converge for the very thin case. Numerical examples are discussed to show that the Mindlin plate element combined with the spline interpolation bases can possess good accuracy.

An Extended Cell-Based Smoothed Three-Node Mindlin Plate Element (XCS-MIN3) for Free Vibration Analysis of Cracked FGM Plates

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) was recently proposed and proven to be robust for static and free vibration analyses of Mindlin plates. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, it is very flexible to apply for arbitrary complicated geometric domains due to using only three-node triangular elements which can be easily generated automatically. However so far, the CS-MIN3 has been only developed for isotropic material and for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-MIN3 by integrating itself with functionally graded material (FGM) and enriched functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed three-node Mindlin plate (XCS-MIN3) for free vibration analysis of cracked FGM plates. Three numerical examples with different conditions are solved and compared with previous published results to illustrate the accuracy and reliability of the XCS-MIN3 for free vibration analysis of cracked FGM plates.

Analyses of Stiffened Plates Resting on Viscoelastic Foundation Subjected to a Moving Load by a Cell-Based Smoothed Triangular Plate Element

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. The CS-MIN3 overcomes shear-locking phenomena and has a faster convergence compared with the original MIN3. In addition, the CS-MIN3 uses only three-node triangular elements that can be easily generated automatically for arbitrary complicated geometric domains. This paper extends the CS-MIN3 by integrating itself with the Timoshenko beam element to give a new stiffened plate element for static, free vibration and dynamic responses of stiffened plates resting on viscoelastic foundation subjected to a moving load. The viscoelastic foundation is modeled by discrete springs and dampers, and the displacement compatible condition between the plate and the stiffener is imposed. Some benchmark numerical examples were performed to illustrate the good agreement of the CS-MIN3 results with those by other methods in the literature to illustrate its accuracy and reliability. In addition, some new numerical examples that consider the effects of stiffeners on the behaviors of plate resting on viscoelastic foundation are conducted. The results demonstrate the expected properties in which the deflection of the stiffened plate resting on viscoelastic foundation can be reduced significantly.

Dual inclined perforated anti-motion plates for mitigating hydroelastic response of a VLFS under wave action

This paper investigates the hydroelastic responses of a mat-like, rectangular very large floating structure (VLFS) edged with dual horizontal/inclined perforated plates using analytic method, numerical method and experimental test. In the analytic method, the eigenfunction expansion-matching method (EEMM) for multiple domains is applied to evaluate the diffraction and radiation potentials, and then the elastic equation of motion is solved by the Rayleigh–Ritz method. In the numerical model, the modal eigenvector equation of the VLFS with some discrete Mindlin plate elements are obtained by using the finite element method (FEM), whereas the boundary element method (BEM) is applied to solve the water wave equation. The hybrid finite element method-boundary element method (FEM-BEM) solutions are further employed for more general cases including inclined perforated anti-motion plates. Both analytic and numerical solutions are also validated against a series of experimental tests. The effectiveness of dual perforated plates in reducing the deflections of VLFS, is systematically assessed for various wave and plate parameters, and the performance of the anti-motion device can be significantly improved by selecting the proper design parameters. Based on the selected optimal parameters, the response reduction of VLFS with different locations of lower perforated plate is further highlighted.

Edge-Based Smoothed Three-Node Mindlin Plate Element

AbstractThe edge-based smoothed finite element method (ES-FEM) was proposed recently to improve the accuracy of the standard finite element method for solid mechanics. In the present paper, the ES-FEM is incorporated with the three-node Mindlin plate element (MIN3) to give a novel edge-based smoothed MIN3 (ES-MIN3) for plate analysis. The system stiffness matrix is computed by employing the edge-based strain smoothing technique over the edge-based smoothing domain. For the purpose of avoiding the transverse shear-locking, the MIN3 element is performed to calculate the strains in each element. From a series of selected numerical examples, it is found that the present ES-MIN3 possesses highly accurate solutions, and can be competitive with many existing plate elements.

Coupled Analysis of Structural–Acoustic Problems Using the Cell-Based Smoothed Three-Node Mindlin Plate Element

The cell-based smoothed finite element method (CS-FEM) using the original three-node Mindlin plate element (MIN3) has recently established competitive advantages for analysis of solid mechanics problems. The three-node configuration of the MIN3 is achieved from the initial, complete quadratic deflection via ‘continuous’ shear edge constraints. In this paper, the proposed CS-FEM-MIN3 is firstly combined with the face-based smoothed finite element method (FS-FEM) to extend the range of application to analyze acoustic fluid–structure interaction problems. As both the CS-FEM and FS-FEM are based on the linear equations, the coupled method is only effective for linear problems. The cell-based smoothed operations are implemented over the two-dimensional (2D) structure domain discretized by triangular elements, while the face-based operations are implemented over the three-dimensional (3D) fluid domain discretized by tetrahedral elements. The gradient smoothing technique can properly soften the stiffness which is overly stiff in the standard FEM model. As a result, the solution accuracy of the coupled system can be significantly improved. Several superior properties of the coupled CS-FEM-MIN3/FS-FEM model are illustrated through a number of numerical examples.

Multiresolution Finite Element Method Based on a New Locking-Free Rectangular Mindlin Plate Element

A locking-free rectangular Mindlin plate element with a new multi-resolution analysis (MRA) is proposed and a new finite element method is hence presented. The MRA framework is formulated out of a mutually nesting displacement subspace sequence whose basis functions are constructed of scaling and shifting on the element domain of basic full node shape function. The basic full node shape function is constructed by extending the split node shape function of a traditional Mindlin plate element to other three quadrants around the coordinate zero point. As a result, a new rational MRA concept together with the resolution level (RL) is constituted for the element. The traditional 4-node rectangular Mindlin plate element and method is a mono-resolution one and also a special case of the proposed element and method. The meshing for the monoresolution plate element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the rigorous mathematical basis. The analysis clarity of a plate structure is actually determined by the RL, not by the mesh. Thus, the accuracy of a plate structural analysis is replaced by the clarity, the irrational MRA by the rational and the mesh model by the RL that is the discretized model by the integrated.

Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification

A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF). It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI) finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD) method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.

Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment

The geometrically nonlinear free vibration of functionally graded thick plates resting on the elastic Pasternak foundation is investigated. The motion equations are derived applying the Hamilton principle. We consider the first order shear deformation plate theory (FSDT), in which the modified shear correction factor is required. A 16-noded Mindlin plate element of the Lagrange family which is free from shear locking due to small thickness of the plate used. The material properties are assumed to be temperature-dependent and expressed as a nonlinear function of temperature. Because the FGM plates are not homogeneous, the basic equations are calculated in the equivalent physical neutral surface which differs from the geometric mid-plane. In the pre-buckling range natural frequencies decrease ultimately reaching zero for critical stress in the bifurcation point.