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PAGE 1 1 FINITE ELEMENT ANALYSIS OF SHELL LIKE STRUCTURES USING IMPLICIT BOUNDARY METHOD By PREM DHEEPAK SALEM PERIYASAMY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Prem Dheepak Salem Periyasamy PAGE 3 3 To Dr. Ramya Baby, Muthuraman, Shardha, Mr. Muthusamy, Mr. Arumugam, Dr. Ashok V.Kumar Dr. Nam Ho Kim Dr. John Kenneth Schueller God, Ravi Burla, Avdhut Joshi, Arjun Ramachandran, Vi vek Raju, t o my roommates and other friends to my other family members and my relatives PAGE 4 4 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor and chairman of my supervisory committee, Dr.Ashok V.Kumar, for his guidance, encourage ment, enthusiasm and constant support throughout my research. I would like to thank him for the numerous insights he provided during every stage of my research. Without his assistance it would not have been possible to complete this dissertation. I would l ike to thank the members of my supervisory committee, Dr. Nam Ho Kim and Dr. John Kenneth Schueller I am grateful for their willingness to serve on my committee, for providing help whenever required, for involvement, for reviewing and valuable suggestions during my thesis. I would also like to thank my colleagues at Design ad Rapid Prototyping Laboratory at the University of Florida for their help and support I would especially thank Ravi Burla, Sung Uk Zhang, Nitin Chandola, Mittu Pannala and Anand Parth asarathy. I would like to thank Ramya Baby, Muthuraman, and Sharadha for their constant love and support. I would like to thank Mr. Muthusamy, Mr. Arumugam for their help and support. Without them this would not have been possible. I would like to thank my friends Avdhut Joshi, Arjun Ramachandran, Vivek Raju and other friends for their constant support Last but not the least; I would like to thank God for giving me this opportunity. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENT S .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 7 LIST OF FIGURES .............................................................................................................................. 8 ABSTRACT ........................................................................................................................................ 11 CHAPTER 1 INTRODUCTION ....................................................................................................................... 13 Overview ...................................................................................................................................... 13 Goals and Objectives .................................................................................................................. 16 Outline .......................................................................................................................................... 16 2 PLATE AND SHELL THEORY ............................................................................................... 18 Plate Theories .............................................................................................................................. 18 Formulation of MindlinReissner Theory .......................................................................... 19 Constitutive Equation for Plate ........................................................................................... 21 Shells as an Assembly of Flat Elements .................................................................................... 23 Stiffness of a Plane Element in Local Coordinates ........................................................... 23 Transformation to Global Coordinates and A ssembly of the Elements ........................... 25 Types of Planar Elements .................................................................................................... 26 Shear Locking .............................................................................................................................. 27 Reissner Mindlin Elements ........................................................................................................ 28 Bathe Dvorkin Element ...................................................................................................... 28 Discrete Kirchhoff Triangle Element (DKT) ..................................................................... 28 Discrete Shear Triangle Element (DST) ............................................................................ 29 3 IMPLICIT BOUNDARY FINITE ELEMENT METHOD ...................................................... 30 Finite Element Method ............................................................................................................... 30 Meshless Methods ....................................................................................................................... 31 Structured Grid Methods ............................................................................................................ 32 Implicit Boundary Finite Element Method (IBFEM) ............................................................... 33 Solution Structure for Imposing Essential Boundary Condition ...................................... 34 Dirichlet Functions .............................................................................................................. 36 Modified Weak Form for Linear Elasticity ........................................................................ 37 Bspline Interpolation .......................................................................................................... 38 One dimensional B spline elements ............................................................................ 39 Two and three dimensional B spline elements ........................................................... 42 Advantages of IBFEM ................................................................................................................ 44 PAGE 6 6 4 SHELLS USING IMPLICIT BOUNDARY METHOD .......................................................... 45 Linear Elastic Problems .............................................................................................................. 45 Formulation of Stiffness Matrix ......................................................................................... 46 Formulation of Stiffness Matrix for Elements with Essential Boundary Conditions ...... 47 Element Force/Load Formulation ....................................................................................... 50 5 ANALYSIS AND RESULTS .................................................................................................... 53 Flat Shell Problems ..................................................................................................................... 53 Cantilever Shell Problem .................................................................................................... 54 Simply Supported Shell Problem ........................................................................................ 57 Centrally Loaded Square Clamped Plate in Bending ........................................................ 60 Obstacle Course Benchmark Problems ...................................................................................... 63 Barrel Vault Roof Problem ................................................................................................. 63 Pinched Hemisphere Problem ............................................................................................. 68 Pinched Cylinder Problem .................................................................................................. 72 Micro Air Vehicle Wing ............................................................................................................. 76 6 CONCLUSION ........................................................................................................................... 80 Conclusions ................................................................................................................................. 80 Scope for Future Work ................................................................................................................ 81 LIST OF REFERENCES ................................................................................................................... 82 BIOGRAPHICAL SKETCH ............................................................................................................. 86 PAGE 7 7 LIST OF TABLES Table page 5 1 Cantilever shell : results for vertical displacement at the end of cantilever shell, based on various meshes and element types ................................................................................... 56 5 2 Simply supported s hell: results for vertical dis placement at the middle of the shell, based on various meshes and element types ......................................................................... 60 5 3 Square p late: results for vertical displacement at the middle of the square plate, based on various meshe s and element types ................................................................................... 63 5 4 Barrel vault roof: results for vertical displacement at the middle of the free edge, based on various meshes and element types ......................................................................... 67 5 5 Pinched hemisphere: results for radial displacement at loaded points of hemisphere, based on various meshes and element types ......................................................................... 71 5 6 Pinched c ylinder: results for vertical displacement at the midpoint of the cylinder, based on various meshes and element types ......................................................................... 75 PAGE 8 8 LIST OF FIGURES Figure page 1 1 Comparison b etween FEM mesh and IBFEM mesh for shells ........................................... 15 2 1 Flat element ............................................................................................................................ 24 2 2 Types of planar elements ....................................................................................................... 27 2 3 Example of a Timoshenko beam ........................................................................................... 27 3 1 Structured grid used in IBFEM ............................................................................................. 34 3 2 Solution structure for imposing essential boundary conditions in IBFEM ........................ 36 3 3 A one dimensional B spline e xample ................................................................................... 40 3 4 Shape function of one dime nsional quadratic B spline element ......................................... 41 3 5 Shape function of one dimensional cubic B spline element ................................................ 42 3 6 A two dimensional quad ratic B spline element in parametric space .................................. 43 3 7 A two dimensional cubic B spline element in parametric space ........................................ 44 4 1 Shells usi ng IBFEM ............................................................................................................... 45 4 2 Representation of narrow band in element with essential boundary condition .................. 49 5 1 Cantilever s hell ....................................................................................................................... 54 5 2 IBFEM result (continuous displacement plot) for cantilever shell example using cubic B spline 64 node elements (10 Elements) .................................................................. 54 5 3 IBFEM result (continuous displacement plot) for cantilever shell example using quadratic B spline 27 node elements (20 Elements) ............................................................ 55 5 4 Abaqus result for cantilever shell example using S4R (25 E lements using continuous displacement plot) .................................................................................................................. 55 5 5 ProMechanica result (continuous displacement plot) for cantilever shell example using shell type model for analysis (unknown mesh/element typ e) ................................... 56 5 6 Simply supported shell ........................................................................................................... 57 5 7 IBFEM result (continuous displacement plot) for simply supported shell example using cub ic B spline 64 node elements (10 Elements) ........................................................ 58 PAGE 9 9 5 8 IBFEM result (continuous displacement plot) for square plate example using quadratic B spline 27 node elements (40 Elements) ............................................................ 58 5 9 Abaqus result for simply supported shell example using shell type model for analysis S4R (900 Elements using continuous displacement plot) ................................................... 59 5 10 ProMechanica result (continuous displacement plot) for simply supported shell example using shell type model for analysis (unknown mesh/element type) .................... 59 5 11 Centrally loaded square plate ................................................................................................ 61 5 12 IBFEM result (continuous displacement plot) for square plate example using cubic Bspline 64 node elements (5 x 5 Mesh) with 0.1% error ................................................... 61 5 13 IBFEM result (continuous displacement plot) for square plate example using quadratic B spline 27 node elements (40 x 40 Mesh) with 0.1% error .............................. 62 5 14 Abaqus result for square plate example using S4R (10 x 10 Mesh) with 0% error (continuous displacement plot) ............................................................................................. 62 5 15 Barrel vault roof ..................................................................................................................... 64 5 1 6 IBFEM result (continuous displacement plot) for barrel vault roof example using cubic B spline 64 node elements (4 x 4 Mesh) with 0.1% error ......................................... 65 5 17 IBFEM result (continuous displacement plot) for barrel vault roof example using quadratic B spline 27 node elements (9 x 9 Mesh) with 0% error ...................................... 65 5 18 Abaqus result (continuous displacement plot) for barrel vault roof example using S 8R5 (9 x 9 Mesh) with 0.6% error (other quarter modeled with different CSYS) from the Abaqus benchmark examples ................................................................................. 66 5 19 ProMechanica result (continuous displacement plot) for barrel vault roof example using shell type model for analysis with 0.9% error (unknown mesh/element type) ........ 66 5 20 Strain energy convergence plot ............................................................................................. 67 5 21 Pinched hemisphere ............................................................................................................... 68 5 22 IBFEM result (continuous displacement plot) for pinched hemisphere example using cubic B spline 64 node elements (20 x 20 Mesh) with 10.3% error ................................... 69 5 23 IBFEM result (continuous displacement plot) for pinched hemisphere example using quadratic B spline 27 node elements (40 x 40 Mesh) with 9.2% error .............................. 69 5 24 Abaqus result (continuous displacement plot) for pinched hemisphere example using S8R5 (8 x 8 Mesh) with 0% error from the Abaqus benchmark examples ........................ 70 PAGE 10 10 5 25 ProMechanic a result (continuous displacement plot) for pinched hemisphere example using shell type model for analysis with 0.1% error (unknown mesh/element type) ................................................................................................................ 70 5 26 Strain energy convergence plot ............................................................................................. 71 5 27 Pinched c ylinder ..................................................................................................................... 72 5 28 IBFEM result (continuous displacement plot) for pinched cylinder example using cubic B spline 64 node elements (20 x 20 Mesh) with 0.2% error ..................................... 73 5 29 IBFEM result (continuous displacement plot) for pinched cylinder example using quadratic B spline 27 node elements (20 x 20 Mesh) with 4.9% error .............................. 73 5 30 Abaqus result (continuous displacement plot) for pinched cylinder example using S4R (20 x 30 Mesh) with 0.4% error from the Abaqus benchmark examples ................... 74 5 31 ProMechanica result (continuous displacement plot) for pinched cylinder example using shell type model for analysis with 0.4% error (unknown mesh/element type) ........ 74 5 32 Strain energy convergence plot ............................................................................................. 75 5 33 Micro air vehicle w ing ........................................................................................................... 76 5 34 IBFEM result (continuous displacement plot) for micro air vehicle wing example using cubic B spline 64 node elements (10 x 5 Mesh) with 1.4% error ............................. 77 5 35 IBFEM result (continuous displacement plot) for micro air vehicle wing exa mple using quadratic B spline 27 node elements (10 x 10 Mesh) with 7% error ....................... 77 5 36 Abaqus result (continuous displacement plot) for micro air vehicle wing example using S8R5 (542 elements) wi th 0.5% error ......................................................................... 78 5 37 ProMechanica result (continuous displacement plot) for micro air vehicle wing example using Shell type model for analysis with 52% error (unknown mesh/element type) ......................................................................................................................................... 78 PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science FINITE ELEMENT ANALYSIS OF SHELL LIKE STRUCTUR ES USING IMPLICIT BOUNDARY METHOD By Prem Dheepak Salem Periyasamy August 2009 Chair: Ashok V. Kumar Major: Mechanical Engineering Shells are structures whose thickness is small compared to their other dimensions. Finite element method (FEM) is the mos t widely used tool for analysis of such structures and s hell elements are used to model such structures. The most basic shell element is a flat element which is formulated based on the Mindlin Reissner theory. The Mindlin Reissner theory yields good result s for thick shells but under predicts deflections for relatively thin shells. In other words, the stiffness of the shell increases as the thickness decreases. This is due to shear locking. Various techniques have been proposed in the past two decades to ci rcumvent this problem and reduced integration is known to be the best solution. FEM uses conforming meshes and automatic mesh generation algorithms can be unreliable for complicated models. In order to avoid the problems associated with mesh generation, se veral meshless methods and structured grid methods have been proposed in the past two decades In this thesis, a structured grid method called implicit boundary finite element method (IBFEM) has been used for analysis of shell like structures. Three dimens ional elements that use uniform B spline approximation schemes for representing the solution are used to model shells. B spline approximations can provide higher order solutions that have tangent and curvature continuity whereas the traditional shell eleme nts use bilinear PAGE 12 12 interpolations which do not even have tangent continuity. T he advantages and disadvantages of using IBFEM and B spline approximation for shell analysis are studied in this thesis. Numerical examples are presented to demonstrate the perform ance of shell elements using IBFEM and B spline approximation. The results are compared with numerically converged solutions and traditional shell elements solutions to demonstrate the ability of the IBFEM shell elements to solve various shell problems in engineering Convergence studies show that shell elements used in this thesis converges with fewer elements and nodes when compared to the 4 node, 8 node and 9 node isoparametric reduced integration shell elements used in FEM. PAGE 13 13 CHAPTER 1 INTRODUCTION Ov erview A shell can be defined as a s tructure with a thickness which is small compared to its other dimensions For example a pressure vessel, whose thickness is less than 1/10 of global structural dimension generally, is m odeled using shell elements. Appli cations of shells include, but are not limited to, structures made using sheet metals fuselages of air planes boat hulls and roof structures in building s The primary difference between a shell structure and plate structure is that, the shell structure s a re subjected to bending and in plane forces (stretching) while the plate structures are subjected to only bending Also the shell structures can be either curved or flat while the plate structures are always flat. The stresses in the thickness directio n ar e negligible for such shell like structures. The finite e lement m ethod (FEM) has been a successful tool for the analysis of shell like structures. Shell theories are extended from the traditional plate theories. There are two plate theories which majorly c ontributed to the advancement of plate and shell analysis. T hey are thin plate theory and thick plate theory. The thin plate theory is based on Kirchhoffs assumption that the rotation at any point in plate is the slope of deflection at the same point i.e ., ;xydwdw dxdy where and xy are the planar axes of plate and w is deflection This theory neglects shear strain energy and requires 1C continuous interpolation or approxima tion of the solution The thick plate theory is based on Mindlins (Mindlin and Reissner 1951) assumption that the deflection and rotation at any point in the plate are independent of each other, i.e., if w (x,y ) is the deflection and ,xy are the rotation about the x and y axis then ,xy are not necessarily the slope of w PAGE 14 14 This assumption allows the plate elements to be 0Ccontinuous because the weak form derived using Mindlins assum ption does not involve second derivatives of the deflection. The most basic shell element is a flat element which is formulated based on the Mindlin plate theory. A shell element should be capable of handling inplane forces as well as bending forces. The 2D plane stress elements can handle in plane forces while the plate elements can handle bending forces. Hence a flat shell element is formulated by combining these two elements (Flugge, 1960) Curved s tructures are approximated by adequate number of flat e lements ( Zienkiewicz 1967) The flat shell element uses linear approximation which is typically 0C continuous. The Mindlin plate theory is good for thick plates but under predicts deflections for relatively thin plates. When the th ickness of the shell reduces the shell based on Mindlin theory tends to stiffen. This is called shear locking in plates and shells (Cook Malkus, Plesha 1989). Various techniques have been proposed to circumvent shear locking. Reduced integration is the best known solution for shear locking (Hughes and Tezduyar, 1981) Increasing the order of the polynomials used for the interpolation also improves the solution (Hartmann and Katz, 2007). In traditional FEM, shell elements are two dimensional flat or curv ed elements that approximate t he geometry as shown in Figure 1 1 (a). In this thesis, analyses of shells using structured grids are studied. A structured grid is a non conforming mesh and all the elements in the grid are regular in shape (rectangles/cuboids ) Figure 1 1 (b) shows a shell modeled using a 3D structured grid where the shell geometry passes through the elements of the grid. A s tructured grid is generated on top of the analysis geometry. The geometry is independent of the grid and could be represe nted using equations or in a format imported from CAD model. PAGE 15 15 A B Figure 1 1 Comparison between FEM mesh and IBFEM mesh for shells. A ) FEM me sh for shell like structures, B ) IBFEM mesh for shell like structures A structured grid is much easier to gen erate than a finite element conforming mesh. Automatic mesh generation algorithms for FEM can be unreliable for complicated geometries often resulting in poor or distorted elements and such distorted elements can cause large errors in solution. Significant amount of user intervention is required which makes mesh generation the most time consuming step in the design process. Moreover, the analysis geometry is approximated using elements. These limitations associated with conforming mesh generation can be avo ided by using a structured grid for analysis which is a significant advantage over traditional FEM. The nodes of a structured grid are not guaranteed to lie on the analysis boundary, and therefore the traditional methods used in FEM for applying essential boundary conditions cannot be used. So a special technique called implicit boundary method (Padmanabhan and Kumar, 2007) is used to enforce essential boundary conditions. Solution structures for the implicit boundary method are constructed by using the app roximate Heaviside step functions based on the implicit equations of the boundaries for exact imposition of essential boundary conditions. The use of structured grids for analysis creates the possibility of using various approximation schemes like B spline s and Hermite interpolations which provide at least 1C continuity in the domain of interest. The following properties of B splines make it desirable to use them for shell PAGE 16 16 analysis. B splines basis functions have compact support lik e finite element shape functions and lead to banded stiffness matrices B spline approximation also provides high order of continuity and is capable of providing accurate solutions with continuous gradients across elements. Burla and Kumar (Burla and Kumar 2008) have used B spline approximation scheme with implicit b oundary finite element m ethod (IBFEM). IBFEM is extended to shell analysis in this thesis. The performance and capabilities of IBFEM shell elements are studied by solving several shell problems in structural analysis. Convergence analysis is also performed using several examples to study the behavior of IBFEM shell elements and it has been demonstrated that IBFEM shell element s provide accurate solutions with a fewer number of elements when comp ared to traditional 4, 8 and 9 node iso parametric reduced integration shell elements Goals and Objectives The goal of this research is to study the shell elements based on B spline approximation using the implicit boundary finite element method and to de velop a solution structure for imposing essential boundary conditions. The main objectives of this thesis are : To extend implicit boundary finite element method to shells and to use B spline approximation for shell analysis To study the convergence behavi or of shell problems using finite elements based on B spline approximations To study the advantage s and disadvantages of using implicit boundary finite element method and B spline approximation for shells Outline The remaining portion of this t hesis is organized as follows: Chapter 2 discusses various traditional plate and shel l theories from literature. V arious assumptions and the limitations such as shear locking are discussed PAGE 17 17 Chapter 3 discusses several meshless methods and methods based on structured g rid for analysis from the literature The i mplicit b oundary finite e lement m ethod and the use of B spline basis functions are discussed in detail Chapter 4 discusses the e xtension of implicit boundary finite element method and B spline approximation to sh ells. The formulation of the solution structure for imposing essential boundary conditions is also presented. S everal shell problems from structural mechanics, obstacle course shell problems and stress concentration problems are studied in chapter 5 Conve rgence analyses are also performed using these shell problems to study the behavior of IBFEM shell elements in terms of accuracy and efficiency In Chapter 6 conclusions are drawn based on the results of the finite element models from chapter 5. Advantage s and disadvantages of using the implicit boundary finite element method and B spline s for shells are also discussed in this chapter. Scope of future work is also presented. PAGE 18 18 CHAPTER 2 PLATE AND SHELL THEORY Thin p lates and shells are structures whose thickness is small compared to their other dimensions The primary difference between a shell and plate is that the shell s are subjected to bending and in plane forces (stretching) while the plate s are subjected to only bending. Also shell s can be either curved or flat while plates are always flat. Applications of thin plates and shells i nclude, but are not limited to, structures made using sheet metals pressure vessels, and fuselages of air planes boat hulls, chimney stacks, automobile parts and roof structures in building. The introduction of thin shells made an important contribution t o the development of several branches of engineering. Some major fields of applications of shell analysis are s tructural engineering v ehicle b ody s tructures a rchitecture and b uildings p ower and c hemical e ngineering and c omposite construction The finite e lement m ethod (FEM) has been a successful tool for the analysis of plates and shells. Since shells elements are extension of plate elements, p late theories and literature are discussed below Plate Theories Some of the greatest contributions towards thin plate theory came from Kirchhoff in 1850. Kirchhoffs assumptions lead to the thin plate theory otherwise known Kirchhoff plate theory. In this theory, the rotation at any point in plate is assumed to be the slope of the deflection at the same point For the plate to remain continuous and not kink, the continuity condition between finite elements has to be imposed to both deflection and rotation ( ;xydwdw dxdy where and xy are the planar axes of plate and w is deflection) The drawback of this theory is that it is valid only for very thin plates where shear strain is negligible. PAGE 19 19 Mindlin and Reissner (Mindlin and Reissner, 1951) developed the thick plate theory. This theory differs from the thin plate theory in basic assumption that the lateral displacement and rotation for any point in a plate are independent. At every node and xyw are treated as the unknown nodal variables where w is deflection and and xy are the rotation s about the x and y axis ( which are not necessarily the slope of w. ). The plate elements based on Mindlins theory are 0 C continuous. This is an advantage of this theory because it is difficult to develop interpola tion that is 1C continuous. Therefore, it is easier to implement thick plate theory than the thin plate theory. The Mindlin plate theory is good for thick plates but under predicts deflections for thin plates. Formulation of Mindli n Reissner Theory Assumptions: For every 2(,), xyAR and 22 tt z 33 30, (,) 1,2 (,) uzxywhere uwxy (2 1 ) In Eq. (2 1), t is the thickness of the plate, 1,2 represents the x and y axis respectively, w is deflection and and xy are rotation s about the z axis T he normal stress in the z direction (33) which is neglected. Weak Form for Linear Elasticity By the principal of virtual work, the weak form for a boundary value in linear elasticity is given as: PAGE 20 20 {}{}{}{}{}{}TTT VSVdVutdSubdV (2 2 ) In Eq. (2 2) {} is the virtual strain, {} u is the virtual displacement, {} is the Cauchy stress tensor {} b is the body force and {} t is the traction Substituting the assumptions (Eq. 2 1) in the weak form for lin ear elasticity ( Eq. 2 2) and simplifying them, we get, the weak form for plate, Weak form for P late: ( )()()AAsKmqdACwFdAMwQds (2 3 ) In Eq. (2 3) 2 2()t tmzdz is the bending moment per unit length, 2 3 2()t tqdz is the shear force per unit length on the face, 2 33 2 t tFbdzt is applied pressure on the plate, 2 2()t tMztdz is pressure (normal stress) on the plate, 2 3 2 t tQtdz is the prescribed bending moment per unit of length, 2 3 2 t tQtdz is the prescribed shear force per unit of length, PAGE 21 21 2 2 t tCzbdzzt is the applied couple per unit of length, t is the sum of shear stress es on the top and bottom surface of the plate Constitutive Equation for P late The c onstitutive equation for plate is sa me as the constitutive equation for plane stress. 11 11 22 22 12 1220 20 00 {}[]{} C (2 4 ) Where 2 1 2(1) EE and are known as Lames constants E is Youngs modulus, is Poisons ratio, {} is the stress vector and {} is the strain vector. S ubstituting the c onstitutive e quation (Eq. 2 4) {}[]{} C in the expression for {m}, we get, 3 22 2 22{}[]{}[]{} []{}[]{} 12tt b ttt mCzdzCkzdzCkDk (2 5 ) In Eq. (2 5), 3[][] 12bt DC and {} q can be re written as {}[]{}sqD (2 6 ) where 0 5 [] 0 6st D t A fter substituting these new expressions for {m} and {q} (Eq. 2 5 and Eq.26) to the LHS of the weak form, the weak form for Mindlin plate is given as Weak Fo rm for Mindlin P late : The weak form for Mindlin plate (Eq. 2 7) is used for analysis of traditional plate elements in FEM. PAGE 22 22 {}[]{}{}[]{} ({}{}{}{})({}{}{}{})TT bs AA TTTT AskDkdADdA CwFdA MwQds (2 7 ) The curvature can be written as, 1 1 1 11 1, 11 2 22 1, 12 2 12 1,1, 12 2100 {} 00 []{} 20x b ye yxw x kN kk N BX x k NN xx (2 8 ) In Eq. (2 8) []bB is the strain displacement matrix for bending and 1 2 k xx 1 1 1,1 1 1 11 1, 1 2 12 2 20 {} []{} 0x s e yw w NN BX NN w (2 9 ) In Eq. (2 9) [] sBis the strain displacement matrix for shear and w x is the shear deformation, [][][][] [][][][]bbbb A ssss AKBDBdA KBDBdA (2 10) In Eq. (2 10) []bK is the stiffness matrix corresponding to bending an d []sK is the stiffness matrix corresponding to the shear, 3[][] 12bt DC and 0 5 [] 0 6st D t PAGE 23 23 Weak form in Matrix Notation: Using Eq. (2 10), the weak form for Mindlin plates in the matrix notation can be writ ten as, {}([][]){}{}([][]) []{}{}TBS TAS eeeeeee eeeXKKXXFK KXF (2 11) In Eq. (2 11) [][][]and{}{}{}BS AS eeeeeeKKKFFF {}A eF is the distributed load and {}s eF is the load contribution from shear. The s tiffness matrix and the force vector for individual elements can be computed and then they can be assembled to solve the equations for unknowns l ike traditional FEM. Shells as an Assembly of Flat Elements The most basic shell element is a flat shell element. A shell element should be capable of handling in plane forces as well as bending forces. The 2D plane stress elements are known to handle in plane forces while the plate elements discussed in the previous section are known to handle bending forces. Hence a flat shell element is formulated by combining these two elements. The formulation for the flat shell element using traditional shell theory from literature is discussed below (Flugge, 1960). Stiffness of a Plane Element in Local Co ordinates Consider a flat element subjected simultaneously to in plane and bending forces. For in plane (plane stress) forces, the state of strain is given by the nodal displacements u and v in x and y direction respectively at node i and we know that, []{}{}{} {}i eppep p i i i p i iu Kxfwithx v U f V (2 12) PAGE 24 24 In Eq. (2 12) []epK is the stiffness matrix corresponding to the plane stress (in plane forces) and U and V are the load components along the x and y axis respectively. Similarly, for bending forces, the state of strain is given by the nodal displacement in z direction ( w ) and the two rotations x and y and we know that, []{}{}{} {}i ebbeb b ixi yi i p ixi yiw Kxfwithx W fM M (2 13) In Eq. (2 13) []ebK is the stiffness matrix corresponding to the bending from Eq. (2 11) while W and M are the load in z direction and moment components along the x and y axis respe ctively. A B Figure 2 1 Flat element. A ) Flat element subjected to in plane forces, B ) Flat element subjected to bending forces Assum p tions: 1 The displacements prescribed for in plane forces do not affect the bending deformations and vice versa 2 For local co ordinates, 0z Now, combining the Eq. (2 12) and Eq. (2 13) we get, PAGE 25 25 []{}{}rs eKxf (2 14) Eq. (2 14) includes stiffness, 0000 0000 000 []000 000 000000p rs b rs rsK KK (2 15) Nodal displacements, {}i i p i i b ii xi zi yi ziu v x w xx (2 16) and respective forces, {}i i i e i xi yi ziU V W f M M M (2 17) The above formulation is valid for any polygo nal element as shown in fig ure 2 1. Transformation to Global Coordinates and Assembly of the Elements A t ransformation of coordinates to a common global system which is denoted by xyz from the local system (denoted by xyz ) is necessary to assemble the elements. The forces and displacements of a node transform from the local to the global system is given by, {}[]{}and{}[]{}iiiixLxfLf (2 18) In Eq. (2 18), 0 [] 0 L (2 19) PAGE 26 26 xxxyxz yxyyyz zxzyzz (2 20) In Eq. ( 2 20), xx cosine of angle between x and x axes, etc By the rules of orthogonal transformation, the stiffness matrix of an element in global co ordinate system can be given as, eTeKTKT (2 21) In Eq. (2 21), 00 00 00 L L T L (2 22) [] T is a matrix made up of [] L matrices along the diagonal equal to the number of nodes in the element. After t he stiffness matrices of all the elements are determined, they are then assembled into a global stiffness matrix and solved for the solution. The resulting displacements correspond to the global coordinate system and have to be transformed to local co ordinate system for the stress computation (Zienkiewicz 1967) Types of Planar E lements 1 Quadrilateral elements: These elements are rectangular in shape and are only used to represent cylindrical or box type s of surface (Zienkiewicz and Cheung, 1965) as shown in the figure 2 2(a). 2 Triangular elements : These elements are triangular in shape and can typically represent any kind of surface as shown in the figure 2 2(b) Although the concept of the use of such elements in analyse s has been suggested as early as 1 961 by Greene, et. al., the success of success of analysis was hampered by the lack of a good stiffness matrix for triangular plate elements in bending (Clough and Tocher, 1965; Clough and Johnson, 1968) Certain limitations of using flat shell elements to approximate the curved geometry are that there is a need for adequate number of elements to represent the smoothness of the shell, the behavior at the inter element boundaries cannot be computed due to discontinuity and the stiffness matrix corresponding to the in plane forces can be ill conditioned. PAGE 27 27 A B Figure 2 2 Types of planar elements. A ) A ssembly of rectangular elements, B ) assembly of triangular elements Shear Locking When the shell thickness shrinks, the Reissner Mindlin plate tends to stif fen. T his problem is primarily due to numerical difficulties If the shell equations could be solved exactly, then if the thickness tends to zero the Reissner Mindlin results should tend to the results of a Kirchhoff plate (Cook Malkus, Plesha 1989). She ar l ocking can be best explained by studying the example of a Timoshenko beam as shown in the figure 2 3. For a short beam, the end deflection is identical to the exact solution. But when the length l is much greater than the height h of the beam, the end deflection will be much too small compared with the exact solution. This is due to shear locking. Figure 2 3 Example of a Timoshenko beam PAGE 28 28 The reason for this stiffening effect is the different sensitivity of the bending stiff ness and the shear stiffness with respect to the height h of the beam. If the height h tend s to zero, the bending stiffness decreases much faster than the shear stiffness. Various techniques have been pro posed to circumvent shear locking and reduced integration is the best known remedy (Hughes and Tezduyar, 1981) Reissner Mindlin Elements Numerous elements are proposed based on Reissner Mindlin plate theory. The three most popular elements are the Bathe D vorkin element, the DKT element and the DST element are discussed in this chapter Bathe Dvorkin Element Bathe Dvorkin element was first developed by Hughes and Tezduyar (Hughes and Tezduyar, 1981) and later ex tended by Bathe and Dvorkin (Bathe and Dvorkin, 1985) to shells The element is an iso parametric four node element with bilinear functions for the deflection w and the rotations x and y The shearing strains are calculated independent of the deflection at the center and at the midpoint of edges Hence the stiffness matrix only depends on the deflections at the four corner points. The main advantage of this element is the easy transition from thick shells to thin shells so that the element is universally applicable. DKT Element The discrete Kirchhoff triangle (DKT ) element can be considered a modified Kirchhoff plate element or a modified Reissner Mindlin plate element ( Stricklin et. al. 1969 ). The element is a triangular el ement The first assumption is that for the rotatio ns, linear functions are chosen and the deflection is instead only defined along the edge and interpolated by Hermite polynomials The second assumption is that there are zero shearing strains at the corne r points PAGE 29 29 and at the midside nodes. The latter assumption makes it possible to couple the rotations to the deflection and it is thereby possible to reduce the model to the three degrees of freedom at the three corner points, so the result is a triangular p late element with the nine degrees of freedom. The DKT element is very popular because with minimal effort ( 0C continuous ) a triangular element with the nodal degrees of freedom w x and y is obtained. But this element is nonconforming (Brae ss, 1997). DST Element DST elements are very similar to DKT elements (Batoz and Katili, 1992). The only difference between DST and DKT elements is that, unlike DKT, the she aring strains are not zero at the corner nodes of the triangle. Although the shell finite elements today can produce good quality results without locking, these results are achieved by using special techniques like reduced integration. Another approach is to increase the order of the polynomials (Hartmann and Katz, 2007). In this thesis, the behavior of quadratic and cubic B spline based 3D shell elements is studied. The results obtained using such shell elements are compared with the traditional shell ele ments. PAGE 30 30 CHAPTER 3 IMPLICIT BOUNDARY FINITE ELEMENT METHO D Finite element method (F EM), various meshless methods and structured grid methods from literature are briefly discussed in the first part of this chapter. A detailed discussion on implicit boundary finite element method (IBFEM) is presented in the second part of this chapter. Ext ension of IBFEM to u niform B spline approximation is presented in the last part of this chapter. Finite Element Method The f inite element method (Hughes, 2000; Cook et. al. 2003) is a well established numerical technique and is widely used in solving engin eering problems in industry, as well as in academia. In finite element analysis the domain of analysis is subdivided into elements and t he resulting finite element mesh approximates the geometry and is also used to approximate the solution by piece wise i nterpolation within each element. Using Galerkins approach, the governing differential equation in its strong form is converted into a weak form which is approximated. This leads to a set of linear equations which can be solved using a sparse matrix solve r, skyline solver, etc. FEM has been applied to wide variety of physical problems in linear/non linear static and dynamic elasticity, steady state heat transfer, electrostatics and magnetostatics. In FEM, a conforming mesh is generated to approximate the shape of the analysis geometry. Automatic mesh generation algorithms for FEM can be unreliable for complicated geometries often resultin g in poor or distorted elements and such distorted elements can cause large errors in the solution. A s ignificant amount of user intervention is required which makes mesh generation the most time consuming step in the design process. In order to avoid these PAGE 31 31 problems with mesh generation, several methods have been proposed which can be classified into two main categories: meshless methods and structured grid methods. Meshless Methods Various methods have been developed in the past two decades which avoid mesh generation and such methods are referred as meshless or meshfree methods. Most of these methods use only nodes that ar e not connected to form elements. Smoothed particle hydrodynamics (SPH) is one of the first meshless methods in literature (Lucy, 1977; Gingold, 1982) This method uses Shepard shape functions as interpolants and uses a point collocation method for discret ization and nodal integration for stiffness matrix computation. The behavior of the solution depends on the proper distribution of nodes. So there is a need to place the nodes properly to avoid instability. Element Free Galerkin Method was developed by Be lytschko and his coworkers (Belytschko et. al., 1994; Dolbow and Belytschko et. al. 1999). This method is based on a moving least square approximation and uses a global Galerkin method for discretization. Boundary conditions were applied using Lagrange mu ltipliers which lead to an increased number of algebraic equations and the stiffness matrix is not symmetric The M eshless Local Petrov Galerkin (MLPG) method was developed by Atluri and his coworkers (Atluri et al. 1999; Atluri and Zhu 2000a, 2000b). Thi s method uses a moving least square approximation and a Local Petrov Galerkin weak form which is developed in the local sub domain. If the sub domain intersects the boundary of the global domain, then the boundary condition is applied on this local boundar y by a penalty method. The a dvantage s of this method are that t here is no need of special integration schemes or smoothing techniques and there is flexibility in choosing the size and shape of the local domain. The drawbacks of this method are that the sti ffness matrices generated by this approach are not symmetric positive definite the PAGE 32 32 penalty method imposes bo undary conditions approximately and there is the need to make sure that the union of all the local domains should cover up the global boundary. T he Partition of unity method (PUFEM) uses a partition of unity functions to merge the local approximation functions which represent the local behavior (Melenk and Babuska, 1996). The main advantage of PUFEM is that the knowledge of the local behavior of so lutions can be used efficiently to get good numerical approximations This method also yields good results in the cases where FEM gives poor results or when FEM is computationally very expensive. One drawback of this approach is that the stiffness matrix c an be singular and the basis function needs to be modified appropriately or special techniques are needed to solve the singular system. The Generalized Finite Element Method (GFEM) was developed as a hybrid of the f inite e lement m ethod and the Partition o f Unity (Babuska et. al. 2002; Duarte et. al. 2000). The desirable features like inter element continuity and the Kronecker delta property from FEM are retained in GFEM. The Natural element method developed by Sukumar ( Sukumar et. al. 1998) uses the natur al neighbor interpolants which are based on a Voronoi tessellation of the nodes. These interpolants are smooth over the whole domain except at the nodes where they are continuous. The essential boundary conditions are applied on nodes for convex domains. S tructured Grid Methods A number of techniques have been developed based on structured grids to approximate the solution. The Extended Finite Element Method or X FEM was developed by Belytschko and co workers (Belytschko et. al. 2003). This method uses imp licit equations for definition of the geometry of the analysis domain and the Dirichlet boundary conditions are applied using Lagrange multipliers. PAGE 33 33 The Penalty boundary method developed by Clark and Anderson (Clark and Anderson, 2003a, 2003b) uses a regul ar structured grid and a constrained variational formulation to apply essential boundary conditions using a penalty method. Shapiro (Shapiro, 1998; Shapiro and Tsukanov, 1999) and Hollig (Hollig et. al. 2001; Hollig and Reif 2003; Hollig 2003) used R func tions to define the boundary of the problem domain and the solution structure proposed by Kantorovich and Krylov (Kantorovich and Krylov 1958) was constructed to satisfy Dirichlet boundary conditions. Shapiro used transfinite Lagrange interpolation and R f unctions to impose Dirichlet boundary conditions while Hollig used weight functions based on distance functions and R functions. Hollig developed extended Bspline basis to represent the solution in the analysis domain. The Implicit boundary finite elemen t method (IBFEM) was developed by Padmanabhan and Kumar (Padmanabhan, 2006 and Kumar et. al. 2007). In this approach, a pproximate Heaviside step functions are used to construct a solution structure based on the technique proposed by Kantorovich and Krylov (Kantorovich and Krylov, 1958). All the internal elements in this method have identical stiffness matrices. Since shell analysis using IBFEM are studied in this thesis, IBFEM is discussed in more detail below. Implicit Boundary Finite Element Method Implic it boundary finite element method (IB FE M) uses structured grid for discretizing the analysis domain as shown in the figure 3 1 T he grid is generated on top of the geometry such that it overlaps the geometry T he geometry is represented independently using equations and is not approximated by the grid. A typical structured grid used in IBFEM is shown in the figure 3 1. Elements which lie completely inside the boundaries of the geometry are called internal elements and the elements which lie partially inside and partially outside the boundary are called PAGE 34 34 the boundary elements. The boundary elements intersect the boundary of the geometry. The nodes are equally spaced in all directions. Figure 3 1 Structured grid used in IBFEM The nodes of a structured gr id are not guaranteed to lie on the analysis boundary. So a special technique called implicit boundary method (Padmanabhan and Kumar, 2007) is used to enforce essential boundary conditions. Solution structures for implicit boundary method are constructed b y using the approximate Heaviside step functions based on the implicit equations of the boundaries for exact imposition of essential boundary conditions. Solution S tructure for I mposing Essential Boundary Condition Essential boundary conditions are typical ly specified as displacements or temperatures on the essential boundaries. Let u be the trial function defined over analysis domain in 2R or 3R The solution structure is constructed such that the essential boundary condition u=a is satisfied in continuum The trial function u based on the technique proposed by Kantorovich and Krylov (Kantorovich and Krylov, 1958) can be given as, uUa (3 1 ) PAGE 35 35 T he trial functi on in Eq. (3 1) is guaranteed to satisfy the boundary condition ua defined by the implicit equation 0 for any : UR The variable part U of the solution structure i s approximated by piece wise polynomial within the elements of the grid Typically, 0 condition is satisfied along the whole boundary of the geometry while the essential boundary conditions are needed to be imposed only along the e ssential boundary Shapiro (Shapiro, 1998; Shapiro and Tsukanov, 1999) used R functions to define the boundary of the problem domain such that takes a value of zero only at the essential boundary. Hollig (Hollig et. al. 2001; Hollig and Reif 2003; Hollig 2003) used distance functions or R functions to define the boundary of the problem domain such that takes a value of zero only at the essential boundary. Both Shapiro and Hollig used B spline basis functio ns to represent the solution in the analysis domain Instead of using any implicit equation Padmanabhan and Kumar (Padmanabhan, 2006 and Kumar et. al. 2007) used Dirichlet function or simply D function which is an approximate step function in the solution structure. The advantage of using approximate step function as D function is that, for all the internal elements, the value of D function becomes unity, and hence the stiffness matrix of all the internal elements are identical. The trial function u is assumed to be displacement in elasticity problems and temperature in heat transfer problems. The trial function u can be approximately represented as, gauDuu (3 2 ) In Eq. (3 2), gu is the grid variable which varies across the elements of the grid au is the boundary value function which becomes ou at essential boundary. A boundary value function is a piecewise continuous function which satisfies the specified values at the essential boundaries. The trail function is g raphically represented in the figure 3 2 PAGE 36 36 Figure 3 2 Solution structure for imposing essential boundary conditions in IBFEM Dirichlet Functions The Dirichlet function is a function whose value becomes zero on all essential boundaries when 0 and when 0 the value increases to unity in a narrow region near the boundary whose width is and then remains unity inside the continuum, In the implicit boundary fi nite element method, the D function is used as an approximate Heaviside step function by considering the limit 0 If the boundary is represented by implicit equation 0 then the step function at any given poi nt x is defined as, 0 0 110 1kD (3 3 ) In the above expressi on, k is the order of D inside F or example, k is 2 for quadratic and 3 for cubic The magnitude of 510 or smaller is used in the implementation When 0 the gr adient of D increases inside the narrow region and is non zero at the essential boundary. This PAGE 37 37 expression is only valid for imposing essential boundary condition on single boundary When a single essential boundary condition has t o be imposed on multiple boundaries 12(,,...,)n passing through same element, then D function has to be constructed as Boolean combination(s) of individual step functions. The Boolean combinations may include union, intersection and subtract ion. For more than two boundaries within one boundary element, a Boolean tree similar to a CSG tree is constructed. Modified Weak Form for Linear Elasticity By the princip l e of virtual work the weak form for a boundary value in linear elasticity is given as: {}{}{}{}{}{}TTT VSVdVutdSubdV (3 4 ) In Eq. (3 3) {} is the virtual strain {} u is virtual displacement, {} is Cauchy stress tensor {} b is body force and {} t is the traction The t rial function is represented as, {}[]{}{}{}{}gasa uDuuuu (3 5 ) In Eq. (3 4) {}[]{}sguDu and [](,...,)dinDdiagDD is a diagonal matrix where iD are D functions that vanish on boundaries on which the thi component of displacement is specified and dn is the dimension of problem, so, {}{}{} {}[]{}[]({}{}){}{}sa sasaCC (3 6 ) Substituting Eq. (3 5) in weak form (Eq. 3 3) we get the modified weak form as {}{}{}{}{}{}{}{}Ts T T TaVSVVdVutdSubdV dV (3 7 ) PAGE 38 38 The nodal values of displacement are approximated as gg ijijuNu wh ere jN are the shape functions and g iju are the nodal values of the grid variable; i represent degree of freedom of field variable and j represents number of nodes The stress and virtual strai n are expressed as {}[][]{}seCBX and {}[]{}eBX respectively, where [] B is the str ain displacement matrix, {}eX and {}eX are the nodal values of the grid variable for element e Using the above expressions, the modified weak form is discretized into a system of linear equations as expressed below, 1 11{}[][][]{}{}{}{}{}{} where, {}{}{}{}{}e eeNE NE NBE eTT e eTe eTT e ee VS eT Ta VVXBCBXdVXFXNtdS FNbdVBdV (3 8 ) In Eq. (3 7) (NE) is the total number of elements in grid and (NBE) is the total number of boundary elements in grid. The element matrices are assembled together to form a global matrix which represents set of linear equations. The left hand sides of the equations correspond to the stiffness matrix and the right hand sides of t he equations correspond to the load vector s Unlike FEM, there is a forcing term in the w eak form which arises from the boundary value functions and also n odal variables to be solved are grid variables and not displacements. Since no assumptions are restricting the choice of shape functions used to represent the grid variables, this approach c an be used with a variety of interpolation schemes such as B spline approximations meshless approximations and Lagrange interpolation. Bspline Interpolation Bspline shape (basis) functions are traditionally constructed using a recursive definition (Far in, 2002). The parameter space is partitioned into elements using a knot vector (equivalent to PAGE 39 39 a collection of nodes). General methods are available to insert knots and elevate the order of the polynomial. IBFEM was recently extended to B spline approxima tion by Burla and Kumar (Burla and Kumar, 2008). In this method, L agrange interpolation is replaced by B spline approximation for representing the solution. A B spline approximation leads to better solution quality and continuity. A B spline interpolation can be classified into 2 types based on polynomial order and they are Quadratic B spline approximation and Cubic B spline approximation Quadratic B spline s are C1 continuous ( gradient continuous) and cubic B spline s are C2 continuous (curvature continuou s) while the conventional finite elements use Lagrange interpolation schemes which are C0 continuous. B spline approximation can also be categorized based on uniformity, and they are uniform B spline s, non uniform B spline s and nonuniform rational B splin es (NURBS) One dimensional Bspline elements The polynomial expressions for the basis functions are derived using the continuity requirements between neighboring elements. A ny thk order B spline has 1 k support nodes. The Bspline polynomials are expressed as, 0 k j iij jNar where ija are the coefficients which determine the continuity requirements, the parameter range for any element is [1,1] r the parameter v a lue for first support node is 1 2 k r and the parameter value for the (1)thk support node is 1 2 k r with uniform parameterization. The piecewise polynomial can be written as, PAGE 40 40 000()kkk j e ii iji iijfrNuaru (3 9 ) In Eq. (3 8), iu represents the values at the support nodes. Th e continuity requirements can be mathematically satisfied using the condition below, 1(1)(1) ,0,1,...,1mm ee mmff mk rr (3 10) In Eq. (3 9) the derivatives of any element e at 1 r are equated to their corresponding derivative s of the (1)the element at 1 r The order of de rivatives limit to 1 k Using this continuity approach, the shape functions for quadratic and cubic B spline s can be derived as shown below. A typical one dimensional B spline is shown in figure 33. Figure 3 3 A one dimensional B spline e xample Quadratic Bspline Element A q uadratic B spline e lement in one dimension has three nodes and therefore it is represented by three shape functions. The expression of the shape functions are given below, PAGE 41 41 2 1 2 2 2 31 (12) 8 1 (62) 8 1 (12) 8 Nrr Nr Nrr (3 11) The plots of the shape functions in Eq. (3 10) are shown in figure 3 4 Figure 3 4 Shape funct ion of one dimensional quadratic B spline element Cubic Bspline Element A c ubic B spline e lement in one dimension has four nodes and therefore it is represented by four shape functions. The expression of the shape functions are given below, 23 1 23 2 23 3 23 41 (133) 48 1 (231533) 48 1 (231533) 48 1 (133) 48 Nrrr Nrrr Nrrr Nrrr (3 12) PAGE 42 42 T he plots of the shape functions in Eq. (3 11) are shown in figure 3 5 This figure shows that the B spline s are not unity at respective nodes and do not vanish at other nodes. This shows that they do not satisfy the kronecker delta property and they also do not interpolate nodal values. Figure 3 5 Shape function of one dimen sional cubic B spline element Two and three dimensional Bspline elements The shape function s for the higher dimensional B spline elements are constructed by taking product of the shape f unctions for one dimensional B spline s. The elements used for grid ar e regular quadrilaterals (rectangle/square) or regular hexahedra (cube/cuboid), so mapping for geometry of elements from parametric space to the physical space is linear. Quadratic Bspline Element The shape functions for two a n d three dimensional quadrat ic B spline s are constructed as a product of one dimensional quadratic B spline s. The shape fun ctions can be expressed as 2 3(1) 3 9(1)3(1)(,)()();,1,2,3 (,,)()()();,,1,2,3D jiij D kji ijkNrsNrNsij NrstNrNsNtijk (3 13) PAGE 43 43 The plots of the shape functions in Eq. (3 12) are shown in figure 3 6 The geometry mapping between parametric and physical space can be defined as below, 11 22lu ii iiirr xxx (3 14) In Eq. (3 13) l ix and u ix are the lower and upper bounds re spectively for nodal coordinates of any given element. Figure 3 6 A t wo dimensional quadratic B spline element in parametric space Cubic Bspline Element The shape functions for two and three dimensional cubic B spline s are constructed by product of one dimensional cubic B spline s. The shape fun ctions can be expressed as 2 4(1) 3 16(1)4(1)(,)()();,1,2,3,4 (,,)()()();,,1,2,3,4D jiij D kji ijkNrsNrNsij NrstNrNsNtijk (3 15) PAGE 44 44 The plots of the shape functions in Eq. (3 14) are shown in figure 3 7 The geometry mapping between parametric and physical space can be defined as the same way as equation (Eq. 3 13) Figure 3 7 A t wo dimensional cubic B spline element in parametric space Advantages of IBFEM A dvantages of using IBFEM for engineering analysis are, The grid generation is simple and quick. The grid is nonconforming and structured and this property saves pre processing time and removes the need for complicated mesh generation algorithms. Use of B spline approximation improves the solution quality and continuity. For example, a problem whose exact solution is a 3rd degree polynomial can be accurately computed using cubic B spline approximation. These are s ome of the i mportant motivating factors to study the behavior of shell elements based on IBFEM using uniform B spline approximation. PAGE 45 45 CHAPTER 4 SHELLS USING IMPLICIT BOUNDARY METHOD The i mplicit b oundary finite element m e thod (IBM) is extended to shells to study the behavior of elements based on B spline approximation for shell analysis The implementation of shell elements is discussed i n detail and the formulation of solution structure for imposing essential boundary conditions is also presented. A B Figure 4 1 Shells using IBFEM. A ) H emispherical solid geometry in IBFEM B) Front vie w of a typical IBFEM grid used for analysis Linear Elastic Problems Shell element s in l inear e lasticity are subjected to both in plane and bending forces A ll the shell elements implemented in this thesis are three dimensional regular hexahedra (cube/cubo id) elements as shown in figure 4 1(b) and (c) The modified weak form for linear elastic problems is discussed in last chapter and can is expressed as, {}{}{}{}{}{}{}{}Ts T T TaVSVVdVutdSubdV dV (4 1 ) From above expression (Eq. 4 1) the modified weak form is discretized into system of equations as expressed below, PAGE 46 46 1 11{}[][][]{}{}{}{}{}{} where, {}{}{}{}{}e eeNE NE NBE eTT e eTe eTT e ee VS eT Ta VVXBCBXdVXFXNtdS FNbdVBdV (4 2 ) The stiffness matrix can be expressed as [][][]eeT VKBCBdV The s tiffness matrix is computed individually for all the elements in grid and then assembled into a global stiffness matrix before solving equations A typical IB FE M grid used for shell analysis is shown in Figure 4 1 Since shells are modeled as open manifolds, there are no internal elements and a ll th e elements in the grid for shell analysis are boundary elements. Amongst all elements only some of the elements contain essential boundary conditions and these elements need a special formulation to impose essential boundary condition s The computation of stiffness matrix for element s with out any boundary condition is discussed first Formulation of Stiffness M atrix The shell geometry typically passes th r ough the element Only a portion of this element that intersects with the volume of the shell contribute s directly to the analysis domain. Therefore special techniques are needed for computation of the stiffness matrix. The portion of shell surface inside any element is subdivided into small triangle s. The stiffness matrix of any element is computed by addin g the stiffness contribution from all the triangles as shown below: 1[][]tn ei iKK (4 3 ) In Eq. (4 3) tn is the total number of triangles in any element e and []iK is the stiffness matrix contribution corresponding to the thi triangle. The stiffness matrix [][][][]iT i VKBCBdV is computed using Gauss q uadrature. Gauss q uadrature is one of the most efficient numerical PAGE 47 47 techniques to evaluate integrals. For shells, t he volume integration is evaluated as a combination of area and thickness integratio n. It is evaluated as follows 2 2[][][][]ih T ii h AKBCBdAdh (4 4 ) In Eq. (4 4) iA is the a rea of thi triangle and h is the thickness of the shell. The normals at the corners of the triangle are known from the model geometry and are interpolated at Gauss points to reduce the cumulativ e error due to triangul ation. Formulation of Stiffness Matrix for Elements with Essential Boundary Conditions For the boundary elements containing essential boundary conditions, the D functions are used and the D functions are defined over a narrow band of width 510 The gradients of D function over the narrow band can have large values. In order to accurately compute the contribution of the D function and its gradient to the stiffness matrix, a special technique is used that is discus sed below The t ria l function in this case is displacement and is represented as, {}[]{}{}{}{}gasa uDuuuu (4 5 ) In Eq. (4 5) {}[]{}sguDu [](,...,)dinDdiagDD is a diagonal matrix where iD are components of D functions and dn is the dimension of problem The nodal values of displace ment are approximated as gg ijijuNu {}[]{}{} {}[]{}s Te eeuDNX BX (4 6 ) In Eq. (4 6) [B] is the strain displacement matrix can be expressed as PAGE 48 48 11 11 11 1232 2 1 22 1112 1121 22110 [] 0 where, is the numberofnodesmND DN xxND B DN xx NDND DNDN xxxx m (4 7 ) 11 11 11 12 32 2 1 22 1112 1211 212100 []0 0mND DN xx ND BDN xx NNDD DDNN xxxx (4 8 ) 11 1 1 1 1 2 1 2 32 2 1 1 2 12 2 12 1 21 10 0 000 0 []000 0 0 00 0mN xD N x D x N D BD N N x DD x DD N xx x (4 9 ) 323442 3222[][][][][] mmm BDNDN (4 10) 12[][][] BBB (4 11) In Eq. (4 11), 1[][][] BDN contains terms involving the gradient of shape functions and 2[][][] BDN contains terms involving the gradient of D function. Now, 12{}[][]{}eeBBX (4 12) PAGE 49 49 Substituting Eq. (4 12) in the stiffness matrix expression 2 2[][][][]ih T ii h AKBCBdAdh 2 1212 2[][][][][][]ih T ii h AKBBCBBdAdh (4 13) Eq. (4 13) can be re written as 2 11122122 2 1223[][][][][][][][][][][][][] [][][][]iiiih TTTT iiiii h AAAA TKBCBdABCBdABCBdABCBdAdh KKKK (4 14) The s tiffness matrix can be decomposed into three main components as shown in equation (Eq. 4 14) Here 1[] K contains ter ms involving only gradients of shape functions, so this component can be computed by subdividing the geometry into triangles. 2[] K and 3[] K contain terms involving gradients of D functions and vanish in the shaded region shown in the figure 4 2 The g radient of D function is very large inside the narrow band and cannot be neglected. Hence 2[] K and 3[] K are computed only in the narrow region from 0 to Figure 4 2. Representation of narrow band in element with essential boundary condition PAGE 50 50 22[][]TKK contain terms involving both gradients of D functions and gradients of shape functions The g radient of shape functions can be assumed to be constant across the band but varies along the boundary while the gradient of D function can be assumed to be constant along the boundary but varies normal to the boundary The boundary of shell geometry is subdivided into li ne segments. Th e second component of stiffness matrix 2[] K can be evaluated as a combination of a line integral along the normal to the boundary and a line integral along the line segment of the boundary and a line integral along the shell thickness Hence 2[] K can be written as, 2 212 2 22 11 [][][][] [][]j lh T jj h l n j jKBCBddldh KK (4 15) In Eq. (4 15) ln is the total number of line segments in the element e and 2[]jK is the stiffness component of thj line segment. Similarly, the last term ca n be written as, 2 322 2 33 11 [][][][] [][]j lh T jj h l n j jKBCBddldh KK (4 16) T he stiffness matrices of all the triangles ar e added to compute the stiffness matrix for its corresponding element. The technique presented above is used to compute the stiffness matrix of boundary elements containing essential boundary conditions. Element Force/ Load F ormulation The discretized form of the modified weak form in Eq. (4 2) is re written below, PAGE 51 51 1 11{}[][][]{}{}{}{}{}{} ,{}{}{}{}{}e eeNE NE NBE eTT e eTe eTT e ee VS eT Ta VVXBCBXdVXFXNtdS whereFNbdVBdV The right hand side of the above expression contains two terms. The term {}eF corresponds to volume integration for the body forces and applied no n homogenous essential boundary conditions and the second term is a surface integration for the traction forces over the natural boundaries. T he shell boundaries are approximated as a set of line segments and are evaluated using Gauss q uadrature as shown b elow, 2 2 1{}{}{} {}i lh T i h L n i ifNtddh ff (4 17) In Eq. (4 17) {} t is the traction acting on the natural boundary, the traction forces are evaluated as combination of line integrals The load is integrated along the boundary iL and throu gh thickness h The traction forces for individual IB FEM shell elements can be computed using the above formulation 2 2{}{}{}h eT h AFNbdAdh (4 18) In Eq. (4 18) the body forces are evaluated as combination of line and area integrals. The body forces are integrated along the surface of the geometry A and through thickness h The body forces for individu al IBFEM shell elements can be computed using the above formulation. PAGE 52 52 Once the stiffness matrices of all the elements are computed individually, they are all assembled to form a global stiffness matrix like traditional FEM and loads are distributed to the n odes of the elements. Finally they are solved using a sparse matrix solver. PAGE 53 53 CHA PTER 5 ANALYSIS AND RESULTS The first part of this chapter consists of examples which are flat plate like structures. The description of the problem, behavior of shell element s convergence studies, comparison of results with existing shell elements and re sult plots are discussed in this chapter. Element b ehavior s for different material properties and various types of loading conditions are also presented in this chapter. These examples are used to help validate the shell element. The elements used are 64 n ode c ubic B spline elements and 27n ode q uadratic B spline elements T he results produced by these high order element s are compared to the results produced by iso parametric 4 node, 8 node and 9 node reduced integration elements used in Abaqus and shell el ements used in ProMechanica The second par t of this chapter consists of three examples which are commonly known as the obstacle course for shell elements suggested by Belytschko (Belytschko, 1986) A good shell example is said to have the ability to han dle in extensional bending modes of deformation, rigid body motion without straining, and complex membrane states of stress. The obstacle course examples are known to challenge all of the capabilities required in a high performance element. Hence the shell elements formulated in this thesis are evaluated using the obstacle course test and the results are studied Flat Shell Problems The flat shell problems presented in this chapter comprises of a c antilever shell example, s impl y s upported shell example and a c entrally l oaded s quare p late in b ending The elements used are 64 node cubic B spline elements and 27 node quadratic B spline elements. PAGE 54 54 Cantilever S hell Problem Figure 5 1 Cantilever s hell A cantilever shell is subjected to uniformly distributed lo ad of 50 Pa (N/m2) while one end of the shell is fixed (cantilever ). The thickness and the material properties of the shell are shown in the figure 5 1 Figure 5 2 IBFEM result (continuous displacement plot) for cantilever shell example using c ubic B spline 64 n ode elements (10 Elements) PAGE 55 55 Figure 5 3 IBFEM result (continuous displacement plot) for cantilever shell example using q uadratic B spline 27 n ode elements (20 Elements) Figure 5 4 Abaqus result for cantilever shell example using S4R (25 Elements using continuous displacement plot) PAGE 56 56 Figure 5 5 ProMechanica result (continuous displacement plot) for c antilever s hell example using s hell type model for analysis (unknown mesh/element type) Table 5 1. Cantilever shell : results for verti cal displacement at the end of cantilever shell based on various meshes and element types Element type Elements Max. Vertical displacement X 10 3 (m) IBFEM 64N 5 1.088 10 1.1 20 1.106 IBFEM 27N 5 0.904 10 1.072 20 1.101 Abaqus S4R 25 1.110 100 1.110 400 1.111 ProMechanica 1.106 PAGE 57 57 The results of various elements are listed in the table and are compared to a numerically converged value of 1.11 mm of displacement at the free end of the cantilever shell The table shows that the results of IBFEM shell elements are converging with a much fewer number of elements when compared to the 4 node iso parametric reduced integration elements suggested by Hughes and Tezduyar (Hughes and Tezduyar, 1981) The element type and number of elements used by Pro Mechanica are unknown but the results are presented for comparison with one more commercial finite element software. Simply Supported Shell Problem Figure 5 6 Simply supported shell A simply supported shell is subjected to uniformly distributed l oad of 50 Pa (N/m2) while the shell is simply supported at its ends The problem description is shown in figure 5 6. PAGE 58 58 Figure 5 7 IBFEM result (continuous displacement plot) for simply supported shell example using c ubic B spline 64 n ode elements (10 El ements) Figure 5 8 IBFEM result (continuous displacement plot) for s quare p late example using q uadratic B spline 27 n ode elements (40 Elements) PAGE 59 59 Figure 5 9 Abaqus result for simply supported shell example using shell type model for analysis S4R (900 Elements using continuous displacement plot) Figure 5 10. ProMechanica result (continuous displacement plot) for s imply s upported s hell example using s hell type model for analysis (unknown mesh/element type) PAGE 60 60 Table 5 2 Simply supported shell : re sults for vertical displacement at the middle of the shell based on various meshes and element types Element type Elements Max. Vertical displacement X 10 4 (m) IBFEM 64N 5 1.175 10 1.187 20 1.187 IBFEM 27N 10 1.133 20 1.182 40 1.187 S4R 100 1.155 400 1.169 900 1.172 ProMechanica 1.174 The results of various elements are listed in the table and are compared to a numerically converged value of 0.1187 mm of displacement at the mid length of the simply supported shell The table shows t hat the results of IBFEM shell elements are converging with a much fewer number of elements when compared to the 4 node isoparametric reduced integration elements suggested by Hughes and Tezduyar (Hughes and Tezduyar, 1981) Centrally Loaded Square Clamp ed Plate in Bending A square plate as shown in the figure 5 11 is loaded centrally by a uniform load of 100 psi. The material used is structural steel. The n ature of the element behavior, convergence studies, comparison of results with existing FEM techniq ues, result plots are presented below. The problem description is shown in figure 5 11. The elements used are 64 node cubic B spline elements and 27 node quadratic B spline elements. PAGE 61 61 Figure 5 11. Centrally l oaded square plate Figure 5 12. IBFEM res ult (continuous displacement plot) for square plate example using c ubic Bspline 64 n ode elements (5 x 5 Mesh) with 0.1% error PAGE 62 62 Figure 5 13. IBFEM result (continuous displacement plot) for square plate example using q uadratic B spline 27 n ode elements (40 x 40 Mesh) with 0.1% error Figure 5 14. Abaqus result for s quare p late example using S4R (10 x 10 Mesh) with 0% error (continuous displacement plot) PAGE 63 63 Table 5 3 Square plate : results for vertical displacement at the middle of the square plate, based on various meshes and element types Element type Mesh Max.Vertical displacement X 103 (in) Error compared to 4.055 X 103 in IBFEM 64N 2 x 2 3.5 62 12.2% 5 5 4.051 0.1% 10 10 4.055 0 .0% IBFEM 27N 10 x 10 3.966 2.2 % 20 20 4.043 0.3 % 40 4 0 4.059 0.1 % Abaqus S4R 5 5 3.703 8.7 % 10 10 4.054 0.0 % 20 x 20 4.055 0.0% The results of various elements are listed in the table and are compared to a numerically converged value of 4.055 x 103 inch of vertical displacement at the middle of the square plate The table shows that the results of IBFEM shell elements are converging with a much fewer number of elements when compared to the 4 node isoparametric reduced integration elements suggested by Hughes and Tezduyar (Hughes and Tezduyar 1981) Obstacle Course Benchmark Problems The s hell o bstacle course comprises of three problems, the b arrel vault roof pinched hemisphere and pinched cylinder. Relevance of these problems in the shell analysis is discussed extensively in the literature by Belytschko (Belytschko, 1986). Barrel Vault R oof P roblem The dimensions of the structure for this problem are shown in the Figure 5 1. The Barrel vault roof is subjected to a gravity load. The ends are fixed to a diaphragm while the side edges are fre e. Using symmetry, only one fourth of the structure has been modeled and meshed The geometric dimensions, material properties, loads and constraints are shown in the figure 5 15. PAGE 64 64 The thickness of the shell is 3 inch The converged numerical solution: vert ical displacement at mid side of the free edge is 3.59 inch ( Ashwell and Gallagher 1976). Figure 5 15. Barrel vault roof The elements used are 64 node cubic B spline elements and 27 node quadratic B spline elements and the results are compared with A baqus 8 node and 9 node isoparametric shell elements. PAGE 65 65 Figure 5 16. IBFEM result (continuous displacement plot) for barrel vault roof example using c ubic B spline 64 n ode elements (4 x 4 Mesh) with 0.1% error Figure 5 17. IBFEM result (continuous d isplacement plot) for barrel vault roof example using q uadratic B spline 27 n ode elements (9 x 9 Mesh) with 0 % error PAGE 66 66 Figure 5 18. Abaqus result (continuous displacement plot) for barrel vault roof example using S8R5 (9 x 9 Mesh) with 0.6% error (other quarter modeled with different CSYS) from the Abaqus b enchmark e xamples Figure 5 19. ProMechanica result (continuous displacement plot) for b arrel vault roof example using s hell type model for analysis with 0.9% error (unknown mesh/element type) PAGE 67 67 Table 5 4 Barrel vault roof : results for vertical displacement at the middle of the free edge, based on various meshes and element types Element type Mesh Vertical displacement Error compared to 91.2 mm (3.59 in) (mm) (in) IBFEM 64N 2 2 9 0.91 3.5 79 0. 3 % 4 4 91.14 3. 588 0.1 % 9 9 92.13 3. 627 1.0 % IBFEM 27N 2 2 109.60 2.466 31.3% 4 4 95.99 3. 350 6.7 % 9 9 92.53 3. 591 0.0 % S8R5 2 2 92.89 3.657 1.9% 4 4 91.74 3.612 0.6% 9 9 91.72 3.611 0.6% S9R5 2 2 92.89 3.657 1.9% 4 4 91.74 3.612 0.6% 9 9 91.72 3.611 0.6% ProMechanica 92.01 3.622 0.9% Figure 5 20. Strain energy convergence plot The results of various elements are listed in the table and are compared to a numerically converged value of 3.59 inch of displac ement at the mid length of the free end ( Ashwell and Gallagher 1976). Figure 5 20 shows that the results of IBFEM shell elements are converging PAGE 68 68 with a much fewer number of elements when compared to the 8 node and 9 node isoparametric reduced integration elements used by Abaqus. Pinched Hemisphere Problem In the pinched hemisphere, the equal and opposite concentrated forces are applied in the anti podal points of the equator ( Abaqus benchmarks manual and NAFEMS b enchmarks 1990), the bottom circumferential edge of the hemisphere is free. Due to symmetry, only one fourth of the structure is modeled and meshed as shown in the figure 5 20. The material properties, modulus of elasticity, E = 6.825e7, Poissons ratio is 0.3. The thickness of the shell is 0.04m The converged numerical solution: radial displacement at loaded points is 185 mm and this quantity is analyzed for convergence (NAFEMS Benchmarks 1990). Figure 5 21. Pinched hemisphere PAGE 69 69 Figure 5 22. IBFEM result (continuous displacement plot) for pinched hemisphere example using c ubic B spline 64 n ode elements (20 x 20 Mesh) with 10.3% error Figure 5 23. IBFEM result (continuous displacement plot) for pinched hemisphere example using q uadratic B spline 27 n ode elements (40 x 40 Mesh) with 9.2% error PAGE 70 70 Figure 5 24. Abaqus result (continuous displacement plot) for pinched hemisphere example using S8R5 (8 x 8 Mesh) with 0% error from the Abaqus b enchmark e xamples Figure 5 25. ProMechanica result (continuous displacement plot) for pinched h emi sphere example using s hell type model for analysis with 0.1% error (unknown mesh/element type) PAGE 71 71 Table 5 5 Pinched hemisphere: results for radial displacement at loaded points of hemisphere, based on various meshes and element types Element type Mesh Disp lacement (m) Error compared to 185 mm IBFEM 64 Node 5 x 5 0.118 36.2% 10 x 10 0.147 20.6 % 20 x 20 0.166 10.3% IBFEM 27 Node 10 x 10 0.121 34.6 % 20 x 20 0.150 18.9 % 40 x 4 0 0.168 9 .2 % S8R5 4 x 4 0.179 3.2 % 8 x 8 0.185 0.0 % S9R5 4 x 4 0. 179 3.2% 8 x 8 0.185 0.0% ProMechanica 0.1853 0.1 % Figure 5 26. Strain energy convergence plot The results of various elements are listed in the table and are compared to a numerically converged value of 185 mm of radial displacement at loaded points of hemisphere (NAFEMS Benchmarks 1990). Figure 5 26 shows that the results of IBFEM shell elements are not very accurate even with a higher number of elements and the results are poorer when compared to the PAGE 72 72 8 node and 9node isoparametric reduced integration elements used in Abaqus. The reason(s) of poor results for this example are still unclear. Pinched Cylinder Problem The pinched c ylinder is subjected to equal and opposite concentrated forces at the mid span and the ends of the cylinder are fix ed to rigid diaphragm. Due to this boundary condition, there is a concentrated stress at the point of loading and this causes localized deformation. Due to symmetry, only one eighth of the structure is modeled and meshed. The geometric dimensions, material properties, loads and constraints are shown in the figure 5 25. The converged numerical solution: radial displacement at loaded points is 1.825 x 105in and this quantity is analyzed for convergence (Lindberg Olson and Cowper 1969) Figure 5 27. Pi nched cylinder The elements used are 64 node cubic B spline elements and 27 node quadratic B spline elements and the results are compared with Abaqus 8 node and 9 node isoparametric shell elements. PAGE 73 73 Figure 5 28. IBFEM result (continuous displacement pl ot) for p inched c ylinder example using c ubic B spline 64 n ode elements (20 x 20 Mesh) with 0.2% error Figure 5 29. IBFEM result (continuous displacement plot) for pinched cylinder example using q uadratic B spline 27 n od e elements (20 x 20 Mesh) with 4. 9 % error PAGE 74 74 Figure 5 30. Abaqus result (continuous displacement plot) for pinched cylinder example using S4R (20 x 30 Mesh) with 0.4% error from the Abaqus b enchmark e xamples Figure 5 31. ProMechanica result (continuous displacement plot) for pinched cylinder example using s hell type model for analysis with 0.4% error (unknown mesh/element type) PAGE 75 75 Table 5 6 Pinched cylinder : results for vertical displacement at the midpoint of the cylinder based on various meshes and element types Element type Mesh D isplacement ( 105 in) Error (compared to 1.825 10 5 in) IBFEM 64 Node 5 x 5 1.537 15.8% 10 x 10 1.768 3.1% 20 x 20 1.822 0.2% IBFEM 27 Node 5 x 5 0.712 61% 10 x 10 1.317 27.8% 20 x 20 1.736 4.9% S8R5 10 x 10 1.804 1.1% 20 x 20 1.833 0.4% S9R5 10 x 10 1.804 1.1% 20 x 20 1.833 0.4% ProMechanica 1.818 0.4% Figure 5 32. Strain energy convergence plot The results of various elements are listed in the table and are compared to a numerically converged value of 1.825 x 105 inc h of vertical displacement at the midpoint of the cylinder (Lindberg, Olson and Cowper 1969) Figure 5 32 shows that the results of IBFEM shell elements are converging but at a much slower rate. This is an example of stress concentration problem and more elements are needed in the areas of stress concentration for better results. But PAGE 76 76 the elements used in this thesis are higher order elements and are computationally expensive when compared to the 8 node and 9 node isoparametric reduced integration elements used by Abaqus. By using adaptive mesh refinement techniques, computational resources can be concentrated near the location of interest (stress concentration) and can be conserved at all other locations. Micro Air Vehicle Wing The dimensions of the struct ure for this problem are shown in the figure 5 30. The micro air vehicle wing is subjected to a traction load. One end is fixed to a diaphragm while the side edges are free. Due to lack of symmetry the whole structure has been modeled and meshed. The geom etric dimensions, material properties, loads and constraints are shown in the figure 5 30. The thickness of the shell is 0.1 inch. The converged numerical solution: vertical displacement at tip of the free edge is 2.1 5 inch and this quantity is analyzed fo r convergence. Figure 5 33. Micro Air Vehicle Wing PAGE 77 77 Figure 5 34. IBFEM result (continuous displacement plot) for micro air vehicle wing example using c ubic B spline 64 n ode elements (10 x 5 Mesh) with 1.4 % error Figure 5 35. IBFEM result (continuous displacement plot) for micro air vehicle wing example using q uadratic B spline 27 n ode elements (10 x 10 Mesh) with 7 % error PAGE 78 7 8 Figure 5 36. Abaqus result (continuous displacement plot) for micro air vehicle wing example using S8 R5 (542 elements ) wit h 0.5 % error Figure 5 37. ProMechanica result (continuous displacement plot) for micro air vehicle wing example using Shell type model for analysis with 52 % error (unknown mesh/element type) PAGE 79 79 The figure s show that the results of IBFEM shell elements ar e consistent with the results of the 8 node and 9node isoparametric reduced integration elements used by Abaqus. For unknown reasons the results of the shell elements used in P ro M echnica are poorer PAGE 80 80 CHAPTER 6 CONCLUSION Conclusions In this thesis, she ll elements based on uniform B spline basis defined over a structured grid are constructed to obtain solutions with continuous gradients throughout the analysis domain. Implicit b oundary f inite e lement m ethod (IBFEM) using B spline approximation is extende d to shells. Both q uadratic and c ubic B spline basis are used for shell analysis. The weak form for linear elasticity was modified such that essential boundary conditions are imposed properly. Methods for efficiently and accurately evaluating the terms inv olved in the modified weak form were developed The primary motivation is the desire to obtain good quality results for shell elements in finite element analysis by using structured grids and uniform B spline approximations Results obtained using IBFEM sh ell elements are compared to traditional shell elements used by commercial FEA applications like A baqus and P ro M echanica. The comparison shows that the results obtained using IBFEM shell elements converges with fewer elements when compared to the 4 node, 8 node and 9node isop arametric elements suggested by Hughes and Tezduyar (Hughes and Tezduyar, 1981). The performances of IBFEM shell elements are evaluated and it passes the obstacle course test except for the pinched hemisphere example The reason(s) f or slow convergence behavior for this particular example is still unknown. For all other examples, t he IBFEM shell elements studied in this thesis exhibited good convergence behavior. The advantage of using IBFEM for developing shell elements is the abilit y to use nonconforming mesh / grid for the analysis Generating a uniform structured grid that encloses the geometry is simpler than generating a conforming mesh because structured grids are independent of the complexity of the geometry. Also all the elem ents used in the analysis are regular and PAGE 81 81 undistorted unlike traditional FEM and hence problems associated with distorted elements are eliminated. All the techniques presented in this thesis were implemented into a JAVA program. Several examples were teste d for the purpose of validating the methods developed. A few of those examples are provided in this thesis and it was observed that the methods are capable of providing solutions that are as good as traditional shell elements Scope for Future Work The IBF EM s hell e lements developed in this thesis are currently used for analysis of linear elastic problems only. The IBFEM shell elements can be extended to non linear analysis like large elastic deformation, elasto plastic deformations, and fracture mechanics The performance of the IBFEM shell elements can be further improved by incorporating adaptive mesh refinement. The stress concentrations problems can be analyzed faster with adaptive mesh refinement C omputational resources can be concentrated near the lo cation of interest (stress concentration) and can be conserved at all other locations. 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The Finite Element Method for Solid and Struc tural Mechanics, Butterworth Heinemann PAGE 86 86 BIOGRAPHICAL SKETCH Prem Dheepak Salem Periyasamy was born in 1986 and brought up in a city called Salem in India. He graduated his high school from S ri Rasi V inayaga (SRV ) High School, Rasipuram, India in 2003. He graduated with a Bachelor of Science in mechanical engineering from Visveswaraya National Institute of Technology, Nagpur, India in May 2007. He then enrolled in the Master of Science P rogram in m echanical and a erospace e ngineering at the University of Florida in fall 2007. His areas of interest include application of finite elem ent analysis, numerical methods, computer aided design continuum mechanics, statistics algorithms and computer applications in the field of mechanical engineering. 