首页 Isogeometric analysis of structural vibrations

Isogeometric analysis of structural vibrations

举报
开通vip

Isogeometric analysis of structural vibrations Abstract 1. Thomas J.R. Hughes’s personal recollections of John H. Argyris appeared in the journal Aircraft Engineering in the mid 1950s [2]. I ordered this book and I was again fascinated by the carefully done illustrations of what seemed to me to be very ...

Isogeometric analysis of structural vibrations
Abstract 1. Thomas J.R. Hughes’s personal recollections of John H. Argyris appeared in the journal Aircraft Engineering in the mid 1950s [2]. I ordered this book and I was again fascinated by the carefully done illustrations of what seemed to me to be very complex structural geometries, certainly more complex than the ones appearing in structural analysis texts of that time. One of the articles in this series contained the famous diptych * Corresponding author. Tel.: +1 512 232 7775; fax: +1 512 232 7508. E-mail address: hughes@ices.utexas.edu (T.J.R. Hughes). Comput. Methods Appl. Mech. Engrg. 195 (2006) 5257–5296 John Argyris was my initial inspiration in science. I first heard of the finite element method in 1967 while I was working at General Dynamics, Electric Boat Division in Groton, Connecticut. Scientific journals were made available to staff and one, in particular, caught my attention, the Aeronautical Journal of the Royal Society which contained short articles in each issue by John Argyris reporting the latest developments in the finite element method. I seem to recall that the terminologies ‘‘matrix displacement method’’ and ‘‘matrix force method’’ were still given preference, but eventually they faded away along with other synonyms for finite elements. I remember the meticulously prepared, elaborate drawings in those papers, a hallmark of John’s scientific work. John was at the peak of his scientific powers, his writing was eloquent, and he was clearly having fun developing and naming new elements and element families, such as ‘‘Lumina,’’ ‘‘Hermes,’’ ‘‘Sheba,’’ etc. I was trying to understand what was going on in this revolutionary new field. A frequent reference in these works was Argyris and Kelsey’s Energy Theorems and Structural Analysis [3], a republication of a series of articles which originally This paper begins with personal recollections of John H. Argyris. The geometrical spirit embodied in Argyris’s work is revived in the sequel in applying the newly developed concept of isogeometric analysis to structural vibration problems. After reviewing some funda- mentals of isogeometric analysis, application is made to several structural models, including rods, thin beams, membranes, and thin plates. Rotationless beam and plate models are utilized as well as three-dimensional solid models. The concept of k-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements. Through the use of nonlinear parameteri- zation, ‘‘optical’’ branches of frequency spectra are eliminated for k-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in p-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed and frequencies and mode shapes are computed and shown to compare favorably with experimental results. � 2006 Elsevier B.V. All rights reserved. Keywords: Isogeometric analysis; Structural vibrations; Invariant frequency spectra; Nonlinear parameterization; Finite elements; Rotationless bending elements; Mass matrices Isogeometric analysis of structural vibrations J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes * Institute for Computational Engineering and Sciences, Computational and Applied Mathematics Chair III, The University of Texas at Austin, 201 East 24th Street, ACES 5.430A, 1 University Station C0200, Austin, TX 78712, USA Received 29 August 2005; received in revised form 29 September 2005; accepted 29 September 2005 ‘‘Let no man ignorant of geometry enter,’’ – inscription above the entry to Plato’s Academy, circa 360 BC www.elsevier.com/locate/cma 0045-7825/$ - see front matter � 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.09.027 expressing the equation-by-equation duality of the force and displacement methods. John clearly had a flair for visual pre- sentation of scientific ideas, and once stated that to do structural analysis well, one needed a sense of its ‘‘geometrical 5258 J.A. Cottrell et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5257–5296 beauty.’’ It is no wonder that John was initially attracted to structural analysis given that it is perhaps the most geometrical of scientific subjects. This spirit is also vividly expressed in his book Modern Fuselage Analysis and the Elastic Aircraft [4]. I also remember a prescient article by Argyris entitled, The Computer Shapes the Theory [1]. It was clear John had pre- cisely discerned the future whereas most others were oblivious, and remained so. It may come as a surprise to many young researchers today but, at that time, many scientific luminaries were fiercely resistant to the use of digital computers, and denigrated any work in which computers played an essential role. For example, the prominent mechanician, Clifford Trues- dell, once gave a lecture in Milan entitled, ‘‘The Computer: Ruin of Science and Threat to Mankind’’ [40]. John had clearly foreseen and embraced the future, so attempts like this to legislate science must have seemed particularly laughable. John’s early work attracted me to the subject but I never met him until the first FENOMECH conference at the Uni- versity of Stuttgart in 1978 [36]. He was very gregarious and friendly when I met him. It was not very long after our first meeting that he invited me to join him and William Prager as co-editor of this journal. Frankly, I was shocked at the time because it was only a few years after I completed my Ph.D. Before beginning my duties as co-editor, William Prager passed away. Thereafter, John, Tinsley Oden and I worked together as the editorial team until John retired. During those years, my relationship with John was always very friendly. We spoke on the telephone about editorial issues and had meetings together with the publisher at scientific conferences. He was always respectful of my ideas and gave me complete autonomy over my editorial activities. I always received Christmas cards from him with personal notes and an occasional letter. A frequent theme was the celebration of our friendship. I have fond memories of him. As I think back about the early days of finite elements, and John Argyris’s work in particular, I am struck by the empha- sis on geometry. This is understandable because geometry is the foundation of analysis and, at the time, there was no com- puter representation of geometry that could support computer based analysis. Geometry was encapsulated in mechanical drawings and these were done with pencils and pens on paper, Mylar and vellum.1 Consequently, computer representation of geometry had to be invented and the finite element description was, first and foremost, simply a geometry. With inspi- ration from the matrix theory of frame structures, the idea of geometric discretization of two- and three-dimensional struc- tures into ‘‘elements’’ was in the air. In fact, this concept seems to have taken hold before there was general agreement as to how to derive element stiffness matrices for even the simplest elements. It is interesting to recall the premise behind the 1964 monograph [22]. Its author, Richard H. Gallagher, another finite element pioneer, set out to determine how stiffness matri- ces for the four-node parallelogram, plane stress element were being calculated at various centers around the world engaged in finite element research. Apparently, no two results were identical. Everyone understood that basic element shapes were necessary to build the computer-based geometrical descriptions. The analytical underpinnings followed. (Indeed, ‘‘The computer shapes the theory!’’) Computer aided design (CAD) came later, in the 1970s and 1980s, and with it came a new and different geometrical description of a structure. CAD and the finite element method have had a difficult relationship. They have never been mar- ried, and attempts at living together have also not been very successful. There seem to be irreconcilable differences in their geometric descriptions. CAD surface geometries frequently have gaps and overlaps which need to be fixed prior to finite element mesh generation. Often, features, such as holes, rivets, and bolts, need to be removed in order to generate meshes. Sometimes, features, such as weld fillets, have to be added in order to avoid physically unrealistic stresses. One might con- clude that the CAD description of geometry is not a convenient basis for finite element analysis but one might also con- clude that the finite element description of geometry, typically consisting of faceted, low-order polynomial elements, is not an adequate description of geometry for modern analysis. It is hard to imagine that analysis, on its own, could re-shape CAD because the latter is a significantly larger business than the former. However, CAD descriptions are also unsuitable for animation and visualization, and the computer gaming industry is significantly larger than CAD. So perhaps some changes may be possible in the ensuing years. Engineering analysis has not yet been impacted by major advances in com- putational geometry but the authors believe this is about to change because the time has come for a better geometrical description than that provided by finite elements. This paper takes a step in this direction. It is inspired by, and dedicated to, the memory of John Argyris. 2. Introduction In Hughes et al. [27] we introduced the concept of isogeometric analysis, which may be viewed as a logical extension of finite element analysis. The objectives of the isogeometric approach were to develop an analysis framework based on func- tions employed in computer aided design (CAD) systems, capable of representing many engineering geometries exactly; to 1 Young engineers may not know what vellum andMylar are. Vellum is a translucent drafting material made from cotton fiber. It is strong, erasable, and provides archival quality. Mylar is the trade name of a translucent polyester film used for drafting. J.A. Cottrell et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5257–5296 5259 employ one, and only one, geometric description for all meshes and all orders of approximation; and to vastly simplify mesh refinement procedures. As a primary tool in the establishment of this new framework for analysis, we selected NURBS (non-uniform rational B-Splines; see, e.g., [38,35]). We found NURBS to possess many interesting properties in analysis and excellent results were attained for problems of linear solid and structural mechanics and linear shells mod- eled as three-dimensional solids, and particularly intriguing results were obtained for the linear advection–diffusion equa- tion. Indeed, it was concluded that isogeometric analysis provided a viable alternative to existing finite element analysis procedures and possessed a number of advantages that might be exploited in various situations. However, isogeometric analysis is in its infancy and much basic work remains to be done to bring these ideas to fruition. A fundamental tenet of isogeometric analysis is to represent geometry as accurately as possible. It was argued in [27] that the faceted nature of finite element geometries could lead to significant errors and difficulties. This is schematically con- veyed in Fig. 1. In order to generate meshes, geometrical simplifications are introduced in finite element analysis. For exam- ple, features such as small holes and fillets are often removed. Stress concentrations produced by holes are then missing, and artificial, non-physical, stress concentrations are induced by the removal of fillets. The stresses at sharp, reentrant cor- ners will be infinite, which makes adaptive mesh refinement strategies meaningless. If the refinement is performed to cap- ture geometrical features in the limit, then tight, automated communication with the geometry definition, typically a CAD file, must exist for the mesh generator and solver. It is rarely the case that this ideal situation is attained in industrial set- tings, which seems to be the reason that automatic, adaptive, refinement procedures have had little industrial penetration despite enormous academic research activity. In isogeometric analysis, the first mesh is designed to represent the exact Fig. 1. Schematic illustration comparing finite element analysis and isogeometric analysis meshes for a bracket. geometry, and subsequent refinements are obtained without further communication with the CAD representation. This idea is dramatized in Fig. 2 in which the question ‘‘What is a circle?’’ is asked rhetorically. In finite element analysis, a circle is an ideal achieved in the limit of mesh refinement (i.e., h-refinement) but never achieved in reality, whereas a circle is achieved exactly for the coarsest mesh in isogeometric analysis, and this exact geometry, and its parameterization, are maintained for all mesh refinements. It is interesting to note that, in the limit, the isogeometric model converges to a poly- nomial representation on each element, but not for any finite mesh. This is the obverse of finite element analysis in which polynomial approximations exist on all meshes, and the circle is the idealized limit. In this paper we initiate the study of the isogeometric analysis methodology in structural vibration analysis. In Section 3 we briefly review the basic concepts of isogeometric analysis. The interested reader is also urged to consult our recent work [27] which presents a more comprehensive introduction. We emphasize the concept of k-refinement, a higher-order proce- dure employing smooth basis functions, which is used repeatedly in the vibration calculations later on. In Sections 4–7 we investigate isogeometric approaches to some simple model problems of structural vibration, including the longitudinal vibration of a rod (and, equivalently, the transverse vibration of a string, or shear beam), the transverse vibration of a thin beam governed by Bernoulli–Euler theory, the transverse vibration of membranes, the transverse vibration of thin plates governed by Poisson–Kirchhoff theory, and the transverse vibration of a thin plate modeled as a three-dimensional elastic solid. In the cases of Bernoulli–Euler beam theory and Poisson–Kirchhoff plate theory, we have employed rotationless for- mulations, an important theme of contemporary research in structural mechanics (see, e.g., [14–16,19,31–34]).2 2 It is interesting to note that the BOSOR (Buckling of Shells of Revolution) codes, which were developed in the 1960s by David Bushnell at the Lockheed Palo Alto Research Laboratory, and still enjoy wide use, also employed a rotationless formulation (see [10,11]). h 0 In the one-dimensional cases we performed numerical analyses of discrete frequency spectra. We were also able to theoretically derive the continuous, limiting spectra and we determined that these spectra are invariant if normalized by the total number of degrees-of-freedom in the model. In other words, one is able to determine a priori the error in frequency for a particular mode from a single function, no matter how many degrees-of-freedom are present in the model. These elementary results are very useful in determining the vibration characteristics of isogeometric models and provide a basis for comparison with standard finite element discretizations. It is well known that, in the case of higher-order finite elements, ‘‘optical’’ branches are present in the spectra (see [8]) and that these are responsible for the large errors in the high-frequency part of the spectrum (see [26]) and contribute to the oscillations (i.e., ‘‘Gibbs phenomena’’) that appear about discontinuities in wave propagation problems. The accurate branch, the so-called ‘‘acoustic’’ branch (see [8]), corresponds to the low-frequency part of the spectrum. In finite element analysis, both acoustic and optical branches are continuous, and the optical branches vitiate a significant portion of the spectrum. In isogeometric analysis, when a linear parameterization of the geometrical mapping from the patch to its image in physical space is employed, only a finite number of frequencies constitute the optical branch. The number of modes comprising the optical branch is constant once the order of approximation is set, independent of the number of elements, but increases with order. In this case, almost the entire spectrum corresponds to the acoustic branch. A linear parameterization of the mapping requires a non-uniform distribution of control points. Our previous work [27] describes the algorithm which locates control points to attain a linear parameterization. Spacing control points uni- formly produces a nonlinear parameterization of the mapping. In this case, remarkably, the optical branch is entirely eliminated! The convergence rates of higher-order finite elements and isogeometric elements constructed by k-refinement are the same for the same order basis, but the overall accuracy of the spectrum is much greater for isogeometric ele- ments. These observations corroborate the speculation that the k-method would be a more accurate and economical procedure than p-method finite elements in vibration analysis of structural members. Studies of membranes and thin … Isogeometric Analysis … Fig. 2. ‘‘What is a circle?’’ In finite element analysis it is an idealization attained in the limit of mesh refinement but never for any finite mesh. In isogeometric analysis, the same exact geometry and parameterization are maintained for all meshes. Finite Element Analysis 5260 J.A. Cottrell et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5257–5296 plates provide additional corroboration. We also present some initial studies of mass lumping within the isogeometric approach. The ‘‘row sum’’ technique is employed (see [26]). Due to the pointwise non-negativity of B-Spline and NURBS bases, the row sum technique is guaranteed to produce positive lumped masses but only second-order accurate frequencies are obtained, independently of the order of basis functions employed. This is unsatisfactory but we conjec- ture that, by appropriately locating knots and control points, higher-order-accurate lumping procedures may exist. This is a topic requiring further research. In Section 8, we apply the isogeometric approach to the NASA Aluminum Testbed Cylinder (ATC) which has been extensively studied experimentally to determine its vibration characteristics. Our isogeometric model is an exact three- dimensional version of the ‘‘as-drawn’’ geometry. All fine-scale features of the geometry, such as fillets, are precisely accounted for. Comparisons are made between the experimental data and the numerical results. In Section 9 we draw conclusions. Appendix A presents analytical and numerical results concerning the order of accu- racy of consistent and lumped mass schemes. 3. A brief summary of NURBS-based isogeometric analysis Non-uniform rational B-Splines (NURBS) are a standard tool for describing and modeling curves and surfaces in com- puter aided design and computer graphics (see [35,38]). The aim of this section is to introduce them briefly and to present an overview of isogeometric analysis, for which an extensive account has been given in Hughes et al. [27]. 3.1. B-Splines B-Splines are piecewise polynomial curves composed of linear combinations of B-Spline basis functions. The coefficients are points in space, referred to as control points. 3.1.1. Knot vectors A knot vector, N, is a set of non-decreasing real numbers representing coordinates in the parametric space of the curve: N ¼ fn1; . . . ; nnþpþ1g; ð1Þ where p is the order of the B-Spline and n is the number of basis functions (and control points) necessary to describe it. The interval [n1,nn+p+1] is called a patch. A knot vector is said to be uniform if its knots are uniformly spaced and non-uniform otherwise. Moreover, a knot vector is said to be open if its first and last k
本文档为【Isogeometric analysis of structural vibrations】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑, 图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。
下载需要: 免费 已有0 人下载
最新资料
资料动态
专题动态
is_506731
暂无简介~
格式:pdf
大小:2MB
软件:PDF阅读器
页数:40
分类:工学
上传时间:2011-10-15
浏览量:28