Elsevier

Computer-Aided Design

Volume 82, January 2017, Pages 127-139
Computer-Aided Design

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

https://doi.org/10.1016/j.cad.2016.04.006Get rights and content

Highlights

  • Isogeometric method for free vibration of non-conforming Mindlin plate is proposed.

  • Cases of multi-interface and multi-patches sharing one common point are considered.

  • The mode frequencies of non-conforming Mindlin plate could be predicted.

  • Nitsche based isogeometric method can provide approving convergence.

Abstract

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.

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 x-axis and y-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 ΩR3 is divided into two bodies Ω1 and Ω2, except common internal boundary Γ whose boundaries can be split into two parts, Dirichlet boundary ΓD and Neumann boundary ΓN. The normal unit vector, nm is defined as positive outwards from internal boundary Γ. We accordingly define Ω=Ω1Ω2and

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, Ξ={ξ1,ξ2,,ξn+p+1}, B-spline basis functions are defined in a recursive form as: Ni,0(ξ)={1ifξiξξi+10otherwise for p=0 and Ni,p(ξ)=ξξiξi+pξiNi,p1(ξ)+ξi+p+1ξξi+p+1ξi+1Ni+1,p1(ξ) for p=1,2,3,, where ξi

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).

References (50)

  • L.V. Tran et al.

    Isogeometric analysis of functionally graded plates using higher-order shear deformation theory

    Composites B

    (2013)
  • H. Nguyen Xuan et al.

    Isogeometric analysis of functionally graded plates using a refined plate theory

    Composites B

    (2014)
  • A. Reali et al.

    An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates

    Comput Methods Appl Mech Engrg

    (2015)
  • J. Kiendl et al.

    Isogeometric collocation methods for the Reissner–Mindlin plate problem

    Comput Methods Appl Mech Engrg

    (2015)
  • J. Deng et al.

    Polynomial splines over hierarchical T-meshes

    Graph Models

    (2008)
  • X. Li et al.

    On linear independence of T-spline blending functions

    Comput Aided Geom Design

    (2012)
  • T. Dokken et al.

    Polynomial splines over locally refined box-partitions

    Comput Aided Geom Design

    (2013)
  • Y. Guo et al.

    Nitsche’s method for a coupling of isogeometric thin shells and blended shell structures

    Comput Methods Appl Mech Engrg

    (2015)
  • X. Du et al.

    Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches

    Comput Aided Geom Design

    (2015)
  • C. Annavarapu et al.

    A robust Nitsches formulation for interface problems

    Comput Methods Appl Mech Engrg

    (2012)
  • C. Hesch et al.

    Isogeometric analysis and domain decomposition methods

    Comput Methods Appl Mech Engrg

    (2012)
  • E. Brivadis et al.

    Isogeometric mortar methods

    Comput Methods Appl Mech Engrg

    (2015)
  • D. Nardini et al.

    A new approach to free vibration analysis using boundary elements

    Appl Math Model

    (1983)
  • G. Liu et al.

    An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids

    J Sound Vib

    (2009)
  • A. Leissa

    Vibration of plates

    (1969)
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