ON BICUBIC B-SPLINE METHOD AND ITS APPLICATIONS TO STRUCTURAL DYNAMICS SI WEIJIAN NATIONAL UNIVERSITY OF SINGAPORE 2003 ON BICUBIC B-SPLINE METHOD AND ITS APPLICATIONS TO STRUCTURAL DYNAMICS SI WEIJIAN A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 ACKNOWLEDGEMENTS I wish to express my heartfelt gratitude to my supervisors, Prof Lam Khin Yong and Dr Gong Shi Wei I will be forever grateful for the way they always helped me think about research from a wider perspective; and for all their down-to-earth advice and encouragement I also want to express my thanks for Dr Zong Zhi, Dr Ng Teng Yong and Dr Li Hua From them I learned a tremendous amount about both doing research and presenting it I also thank Ms Zhang Yingyan, Mr Zhang Jian, Mr Yew Yong Kin, Mr Chen Jun and Mr Wang Zijie, with whom I shared my research experience in the Institute of High Performance Computing I would like to express my thanks to the lab mates of the Dynamic/Vibration Lab, including Mr Lu Feng, Mr Khun Min Swe, Mr Tao Qian and Ms Zeng Ying, for their friendships and for all they have done for me I am deeply indebted to my dear wife, Guxiang, for her love and understanding through my graduate years She was always behind me and gave her unconditional support even if that meant to sacrifice the time we spent together i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii ABSTRACT v LIST OF FIGURES vii LIST OF TABLES viii LIST OF NOTATIONS xi Chapter 1 INTRODUCTION 1.1 Introduction 1.2 Application of B-spline Functions to Numerical Analysis 1.3 Review of Vibration Analysis of Plates Using B-spline Functions 1.4 Chapter Arrangement of the Thesis THE NEW FORM OF B-SPLINE BASIS SETS 11 2.1 Introduction 11 2.2 Description of B-splines 12 2.3 Review of Two Forms of B-spline Basis Sets 17 2.3.1 The Form by Qin (1985) 17 2.3.2 The Form by Yuen (1999) 18 2.4 Why the New Form? 20 2.5 Beam Function Approximation 22 ii Chapter 2.5.1 Beams Under the Three Classical B.C.s 22 2.5.2 Beams with at Least One Guided Edge 24 VIBRATION ANALYSIS OF ORTHOTROPIC PLATES 26 3.1 Introduction 26 3.2 Review of Vibration of Rectangular Plates 27 3.3 Bicubic B-spline Approximation 31 3.4 Frequency Equation Formulation 35 3.5 Imposition of Boundary Conditions 37 Chapter NUMERICAL RESULTS FOR VIBRATION OF ORTHOTROPIC PLATES 41 4.1 Introduction 41 4.2 Program Design 42 4.3 Convergence Study 45 4.4 Validation of Bicubic B-spline Method 48 4.4.1 C-C-S-S Isotropic Plate 48 4.4.2 Completely Free Isotropic Plate 48 4.4.3 Fundamental Frequencies of Six Cases of Orthotropic 50 Plates 4.4.4 Five Cases of Orthotropic Plates with Different Boundary 52 Conditions 4.5 Chapter Numerical Results by the Present Method VIBRATION OF PLATES WITH GUIDED EDGE(S) 55 62 5.1 Introduction 62 5.2 Imposition of Boundary Condition of Guided Edge 64 iii 5.3 Comparison with Exact Solutions 65 5.4 Numerical Results by the Present Method 67 5.5 Discussions 74 5.5.1 Meshfree feature of the bicubic B-spline method 74 5.5.2 Advantages of the present bicubic B-spline method 75 Chapter BICUBIC B-SPLINE METHOD FOR TRANSIENT 77 ANALYSIS 6.1 Introduction 77 6.2 Dynamic Equation 78 6.3 Newmark-beta Integration 79 6.4 Numerical Example 81 6.5 Discussion on the Efficiency of the Present Method 83 Chapter CONCLUSIONS AND RECOMMENDATIONS 84 7.1 Conclusions 84 7.2 Recommendations 85 APPENDIX A Notes on Kronecker product 87 APPENDIX B Elements of the Spline Matrices 90 REFERENCES 92 iv ABSTRACT The present thesis focuses on the development of bicubic B-spline method in combination with a new form of B-spline basis set for dynamics analysis of structures A new form of B-spline basis set is first proposed to enrich greater generality of the application of B-spline functions to numerical analysis The development of the new form of cubic B-spline basis set is of considerable significance, since it is much more versatile than are other kinds of beam functions or polynomials with regard to the variety of end conditions that can be accommodated, and there’s no problem in meeting the end conditions for guided edge Using the new form of B-spline basis set, bicubic B-spline approximation procedure is developed for vibration analysis of orthotropic plates The plate deflection is approximated by the product of the new form of B-spline basis set in both x- and ydirections The frequency characteristic equation is derived based on classical thin plate theory by performing Hamilton's principle Various boundary conditions can be handled in this method, and furthermore, in this method the imposition of boundary conditions is very simple A general unified Fortran computer program capable of analyzing vibration of orthotropic as well as isotropic plates under any combination of the four kinds of boundary conditions is developed There exist 36 combinations of the three classical v edge conditions, i.e simply supported, clamped, and free edge conditions, for rectangular orthotropic plates The natural frequencies of orthotropic plates with all 36 combinations of the three classical boundary conditions with various aspect ratios are presented Comparisons with exact solutions and other numerical results demonstrate fast convergence, high accuracy, versatility, and computation efficiency of the present approach In addition to the three classical edge conditions (i.e simply supported, clamped, and free edges), the fourth mathematically possible boundary condition has been referred to in the literature as the guided edge The number of all possible combinations is 34 when at least one guided edge is involved Of all these 34 cases, analytical solution is possible for 21 cases only The solutions of the remaining 13 cases are possible by approximate or numerical methods only, however, no investigation has been reported To show the versatility of the present method, the results for rectangular plates with at least one guided edge are also computed The present method results in a significant reduction in degrees of freedom compared to conventional FEM, which is desirable for dynamic analysis of complex structures Linear transient analysis is also carried out for plate structures and a simple example is provided to shown the high effectiveness of the present method vi LIST OF FIGURES Figure 2.1 Standard cubic B-spline function 13 Figure 2.2 Beam function approximation by ordinary cubic B-spline 15 functions Figure 3.1 Plate deflection approximation by B-spline basis set 32 ( x − direction) Figure 4.1 Flowchart of the program 44 Figure 5.1 Guided (or sliding) edge 63 Figure 6.1 Step load acting on the structure 82 Figure 6.2 Response of a simply-supported plate to step load 82 vii LIST OF TABLES Table 4.1 Boundary indices and corresponding boundary conditions 43 Table 4.2 Convergence of frequency coefficient Ω = ωa ρ D for S-S-S-S 46 isotropic square plate (* double eigenfrequencies) Table 4.3 Convergence of frequency coefficient Ω = ωa ρ D y for clamped 47 orthotropic square plate Table 4.4 Frequency coefficient Ω = ωa ρ D for C-C-S-S rectangular plate 48 with different aspect ratios Table 4.5 Comparison of Natural Frequencies (Hz) of Completely Free 49 Orthotropic Plate Table 4.6 Fundamental frequency coefficient p11 a ρh D y for rectangular 51 plywood plates with different boundary conditions Table 4.7 Comparison of exact and approximate frequencies of S-F-S-C plywood 51 plate Table 4.8 Dimensionless natural frequencies for the E-glass/epoxy rectangular 53 C-S-C-S plate with different aspect ratios Table 4.9 Dimensionless natural frequencies for the E-glass/epoxy rectangular 53 C-S-S-S plate with different aspect ratios Table 4.10 Dimensionless natural frequencies for the E-glass/epoxy rectangular 54 viii encouraging to expect that high versatility of the present method will be demonstrated 2) A logical extension is to investigate the vibration characteristics of shallow shells To this, the transverse as well as in-plane displacement should be approximated using the new form of cubic B-spline basis set And the frequency equation of shallow shell can be formulated by performing Hamilton’s principle It’s anticipated that the present method is capable of solving vibration shell with arbitrary combination of the four kinds of edge conditions 3) In addition to the traditional problems in structural numerical analysis, the present method can be used in computational atomic and molecular physics, for example, multiphoton two-electron ionization It’s a fascinating aspect of the three-body problem, in which the handling of very large and dense basis sets will be required However, splines have the property of being ‘complete enough’ with a relatively small number of basis functions thus it is possible for such a basis set to compete with the finite-difference methods For a detailed reference of the application of B-splines in atomic and molecular physics, the reader is referred to Bachau et al (2001) 86 APPENDIX A Notes on Kronecker Product A1 The Kronecker Product The Kronecker product is a binary matrix operator that maps two arbitrarily dimensioned matrices into a larger matrix with special block structure Given the n × m matrix An×m and the p × q matrix B p×q  a1,1 L a1,m    A= M O M  a n ,1 L an ,m  n×m  b1,1 L b1,q    B= M O M  b p ,1 L b p ,q    p× q their Kronecker product, denoted A ⊗ B , is the np × mq matrix with the block structure  a1,1 B L a1,m B    A⊗ B =  M O M  a n ,1 B L an ,m B  np×mq 87 For example, given 1  A=  0 − 1 2×  3 B=  4 6 2×3 the Kronecker product A ⊗ B is 1 4 A⊗ B =  0  0 6 10 12  0 − − − 3  0 − − − 6 x A2 Properties of the Kronecker Product Operator In the following it is assumed that A, B, C, and D are real valued matrices Some identities only hold for appropriately dimensioned matrices The Kronecker product is a bi-linear operator A ⊗ (αB) = α ( A ⊗ B) (αA) ⊗ B = α ( A ⊗ B) Kronecker product distributes over addition ( A + B) ⊗ C = ( A ⊗ C ) + ( B ⊗ C ) A ⊗ ( B + C ) = ( A ⊗ B) + ( A ⊗ C ) The Kronecker product is associative ( A ⊗ B) ⊗ C = A ⊗ ( B ⊗ C ) 88 The Kronecker product is not in general commutative, i.e usually A⊗ B ≠ B⊗ A Transpose distributes over the Kronecker product (does not invert order) ( A ⊗ B ) T = AT ⊗ B T Matrix multiplication, when dimensions are appropriate, ( A ⊗ B)(C ⊗ D) = ( AC ⊗ BD) When A and B are square and full rank ( A ⊗ B) −1 = A −1 ⊗ B −1 The determinant of a Kronecker product is (note right hand side exponents) det( An×n ⊗ Bm×m ) = det( A) m • det( B) n The trace of a Kronecker product is trace( A ⊗ B) = trace( A) • trace( B) 89 APPENDIX B Elements of the Spline Matrices The elements of the spline matrices [ Ax ], [ B x ], [C x ] , and [ Fx ] in equations (3.2427) are as following The dimension of the spline matrices [ Ax ], [ B x ], [C x ] , and [ Fx ] is ( N + 3) × ( N + 3) The spline matrices [ Ay ], [ B y ], [C y ] , and [ Fy ] are in similar form as [ Ax ], [ B x ], [C x ] , and [ Fx ] , and the dimension is ( M + 3) × ( M + 3)   36 54 15 − 48 6   96 27 − 60   h  [ Ax ] = x  18 − 16 − 36   96 54    Sym O O O O ( N +3)×( N +3) − 396 − 954 − 57 240 − 234 − 624 − 165 84   [Bx ] = − 54 720hx  − 480   Sym     47  90 144  O O O O ( N +3)×( N +3) 150 144 6 90  396 234 57 − 240 − 150 −    624 165 − 84 − 144 −     [C x ] = −8 − 47 −2 54 720hx   − − − 480 90 144    Sym O O O O ( N +3)×( N +3) 36756 19854 5055 7872 726    13776 3819 7140 720   hx   [ Fx ] = 1098 2264 239 30240   14496 7146 720    Sym O O O O ( N +3)×( N +3) 91 REFERENCES Ahmadian, M.T and Sherafati Zangeneh, M Vibration analysis of orthotropic rectangular plates using superelements Computer Methods in Applied Mechanics and Engineering, Vol 191, pp.2097-2103 2002 Alexander Graham Kronecker Products and Matrix Calculus With Applications Halsted Press, John Wiley & Sons, New 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Table 5.4 Comparison of frequency parameters for GGGG plate 67 Table 5.5 Bicubic < /b> B- spline solutions of frequency parameters for CCCG plate 68 Table 5.6 Bicubic < /b> B- spline solutions of frequency parameters for CCSG plate 68 Table 5.7 Bicubic < /b> B- spline solutions of frequency parameters for CCGF plate 69 ix Table 5.8 Bicubic < /b> B- spline solutions of frequency parameters for CCGG plate 69 Table 5.9 Bicubic < /b> B- spline. .. solutions of frequency parameters for CGCF plate 70 Table 5.10 Bicubic < /b> B- spline solutions of frequency parameters for CGSF plate 70 Table 5.11 Bicubic < /b> B- spline solutions of frequency parameters for CSGF plate 71 Table 5.12 Bicubic < /b> B- spline solutions of frequency parameters for CGGF plate 71 Table 5.13 Bicubic < /b> B- spline solutions of frequency parameters for CFGF plate 72 Table 5.14 Bicubic < /b> B- spline solutions... annular sector plates (Mizusawa 1991a), and stepped annular sector plates (Mizusawa 199 1b) , vibration and buckling of plates with mixed boundary conditions (Mizusawa and Leonard 1990), were analyzed by Mizusawa and his colleagues by using spline strip method Both the spline finite strip method and the spline strip method employ cubic B- spline functions in one direction only To fully exploit the desirable... at the breakpoints Cubic B- spline function has advantageous properties in numerical analysis Firstly, cubic B- spline function has non-zero values only over four adjacent sections of the domain, which makes cubic B- spline approximation a local approximation scheme At any spline node, only three terms of B- spline functions have non-zero contributions to the approximation function Secondly, only one degree... characteristics of B- spline function, it’s reasonable to express the displacement function in two directions rather than in one The approximation technique using Bspline functions in two directions is called bicubic < /b> B- spline approximation The mathematical concept of bicubic < /b> spline interpolation was first proposed by Boor (1962) This interpolation technique was originally used in approximation problems, such... investigated by Bert and Malik (1994) The solutions of the remaining 13 cases are possible by approximate or numerical methods only, however, no investigation has been reported In the present thesis, a novel method, called bicubic < /b> B- spline method is developed to solve plate vibration problem under any combination of the four kinds boundary conditions The enforcement of boundary conditions has been one of... clamped, and free boundary conditions 9 Chapter 5 gives the numerical results for free vibration analysis of rectangular plates with at least one guided edge Chapter 6 applies bicubic < /b> B- spline method for transient response analysis in conjunction with Newmark Beta method, simple example is provided to demonstrate the effectiveness and accuracy of the bicubic < /b> B- spline method Conclusions and recommendations... plate 72 Table 5.15 Bicubic < /b> B- spline solutions of frequency parameters for CGFF plate 73 Table 5.16 Bicubic < /b> B- spline solutions of frequency parameters for GGFF plate 73 Table 5.17 Bicubic < /b> B- spline solutions of frequency parameters for GFFF plate 74 Table 5.18 Comparison of total DOF numbers of different methods 74 x LIST OF NOTATIONS a = plate length in x-direction; b = plate breadth in y-direction; [ Ax... is needed to achieve C2–continuity between adjacent sections, whereas cubic Lagrange function and cubic Hermite function have only C0–continuity and C1– continuity, respectively Thirdly, cubic B- spline function can be physically regarded as the deflection function of the beam fixed at both ends In this sense, cubic B- spline function can attain the best interpolation of any given function 14 Consider... at the i spline node For arbitrary point, other than the spline nodes, only four non-zero terms have contributions to the displacement interpolation Thus, cubic B- spline approximation is a local approximation scheme For operation on < /b> the beam function in equation (2.3), the boundary condition at x = 0 and x = l must be considered However, it's difficult to satisfy different boundary conditions using ... response analysis in conjunction with Newmark Beta method, simple example is provided to demonstrate the effectiveness and accuracy of the bicubic B-spline method Conclusions and recommendations... B-spline approximation a local approximation scheme At any spline node, only three terms of B-spline functions have non-zero contributions to the approximation function Secondly, only one degree of... nodes, only four non-zero terms have contributions to the displacement interpolation Thus, cubic B-spline approximation is a local approximation scheme For operation on the beam function in equation