Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch
Introduction
The plate structures are extensively used in industry products such as automobiles, aircrafts, vessels. The vibration of plate plays a vital role in research on the stability of the moving parts, which will influence the security of the structures. Most solutions of natural frequencies are obtained based on classical plate theory, known as Kirchhoff plate theory, which ignores the influence of transverse shear deformation and the rotary inertia terms, and has been popularly used to solve the problems of thin plate [1], [2]. While the neglected terms are quite significant for moderately thick plates and were considered in Reissner–Mindlin plate theory [3], [4]. Different from Kirchhoff assumption, a cross section is not essential to remain perpendicular to the middle surface during the deformation in Mindlin theory. Moreover, two cross-sectional rotations around the -axis and -axis, are independent besides the deflection. The closed-form solutions for free vibration analysis of Mindlin plate are difficult to obtain and the existing acquired solutions are limited to special cases [5], [6], [7]. Owing to the difficulties of obtaining the analytical solutions, many numerical methods have been proposed and successfully applied to free vibration analysis of Mindlin plates such as pb-2 Rayleigh–Ritz method [8], [9], finite strip method [10], [11], meshless method [12], differential quadrature element method (DQEM) [13], DSC-Ritz method [14], smoothed finite element method (SFEM) [15], etc. The newly invented isogeometric method was also begun to be used in the investigation of Mindlin theory based free vibration problems.
Isogeometric analysis (IGA) proposed by Hughes et al. is supposedly to realize the integration of CAD and classical FEA [16]. The benefits of unified geometry model for CAD & CAE, which is an obvious advantage IGA holds and can improve both precision and efficiency of the analysis make it rapidly exploited in miscellaneous fields including structural mechanics and vibration, fluid structure, electromagnetics, acoustics, phase-field analysis. In the application of NURBS-based isogeometric method to plate problems, Shojaee et al. conducted an investigation into the static and free vibration of the thin plates based on Kirchhoff theory [17], [18]. Veiga et al. [19] and Sang et al. [20] employed the Mindlin theory to study the bending problem and the free vibration of thick plates respectively. The static, free vibration and bulking analysis of functionally graded plates have been analyzed based on the first order shear deformation plate theory (FSDT) [21], higher-order shear deformation theory (HSDT) [22] and a refined plate theory (RPT) [23]. Third order shear deformation theory (TSDT) was introduced to study the static and free vibration of composite plates [24], [25]. Recently, an isogeometric collocation approach has also been developed to solve thin and thick plate structural problems [26], [27].
As the basis function of the geometry and the shape function of the analysis in IGA, NURBS possesses many advantages such as accurate expression of geometry, affine invariance, fast and stable matching algorithms and so on [28]. However, it is impossible to perform local refinement and construct complex geometry with singular NURBS patch. Many local refinable splines have been developed to solve the first problem like T-splines [29], polynomial spline over hierarchical T-meshes (PHT-splines) [30], analysis-suitable T-splines [31], local-refined (LR) B-spline [32]. The latter limitation compels us to build complicated model with multiple patches, which will produce continuity and non-conforming problems [33]. In engineering practice, complex geometries involving thousands of surface entities are ubiquitous such as in automobile panels, aircraft skins, vessel hulls, etc. It is impossible to build conforming boundaries in an efficient way when the model is complex and consists of overwhelming number of trimmed geometries, which makes it difficult to assemble the stiffness matrix in IGA. Due to such difficulties NURBS encounters, Nitsche method, which was originally proposed to weakly impose Dirichlet boundary conditions [34], has been developed as a coupling tool to tackle two and three dimensional linear elasticity problem [35], [36], thin Kirchhoff–Love shell problem [37], Reissner–Mindlin static problem [38] under the framework of NURBS-based isogeometric analysis. However, the non-conforming isogeometric models studied in existing literatures are simple and only one-to-one patch correspondence cases or cases with a minority of interfaces are considered.
In this study, isogeometric method is used to study the free vibration analysis of non-conforming Reissner–Mindlin plate with various model shapes, different boundary conditions and several kinds of thickness-span ratios. Multi-interface problems and several patches sharing one common point are investigated by using Nitsche method to glue different patches. The existing reference solutions and the results computed by ABAQUS software have been compared with the presented results, showing the accuracy, efficiency and convergence of the dimensionless frequency parameters.
The paper is organized as follows. Section 2 introduces the non-conforming multi-domain problems and the weak enforcement of the constrains on coupling conditions with Nitsche method. The NURBS basis function and its derivatives, Reissner–Mindlin plate theory are briefly described in Section 3. Meanwhile, the discrete equations of free vibration are deduced by solving an eigenvalue problem in the framework of Nitsche based isogeometric analysis. Several numerical examples are shown in Section 4. Section 5 draws the conclusion and discusses some challenges and future works.
Section snippets
Problem description
We consider the Reissner–Mindlin plate problem in the context of elastodynamics. Two-domain elasticity boundary value problem (BVP) is shown in Fig. 1 as a general case of multi-domain problem. The whole domain is divided into two bodies and , except common internal boundary whose boundaries can be split into two parts, Dirichlet boundary and Neumann boundary . The normal unit vector, is defined as positive outwards from internal boundary . We accordingly define
NURBS and its derivatives
Non-uniform Rational B-Spline (NURBS) not only serves as a standard tool for modeling geometries [28], but also to approximate the solution field in isogeometric analysis. B-spline and NURBS functions as well as their derivatives are briefly reviewed in this section.
Given a knot vector, , B-spline basis functions are defined in a recursive form as: for and for , where
Numerical tests
In this section, several numerical examples of non-conforming multi-patch plates containing square plate, circular plate, regular hexagonal plate and shear wall plate are presented to verify the performance of isogeometric method. Nitsche approach is employed to glue different patches. The isogeometric meshes are non-conforming along the common internal boundaries. Different boundary conditions containing simply supported (S), Clamped (C) and free (F) were enforced on the plate. Unless
Conclusions
In this study, free vibration analysis of Reissner–Mindlin plate has been successfully performed by using Nitsche method in conjunction with NURBS-based isogeometric method, where Nitsche method is used to glue different NURBS patches and weakly constrains the coupling conditions. The numerical examples of multi-patch plates are analyzed to verify the validity of the Nitsche based isogeometric method. The effects of various geometry shapes, different boundary conditions, and several kinds of
Acknowledgment
The work is supported by the Natural Science Foundation of China (Project Nos. 51305016 and 11402015).
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