On some properties on bivariate Fibonacci and Lucas polynomials

2nd International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 2017) Mathematical Model of Catmull-Clark Subdivision Scheme on RegularMeshShu-qun Liu, Bei ZhangSchool of Computer and Communication, Lanzhou University of Technology, Lanzhou 730050,China*****************Keywords: Subdivision, interpolation, mathematical expression, B-spline surface.Abstract. In order to further simplify the research of limit surface properties and establish a unified mathematical model. According to Catmull-Clark subdivision method in the regular mesh at the subdivision rules, a method of calculating the mathematical expression of limit surface by interpolation was presented by using the subdivision method which is described by the curves and surfaces interpolation theory. And according to the process of solving, we prove that the limit surface which generated by the Catmull-Clark subdivision method in the regular mesh is a bi-cubic B-spline surface.1. IntroductionThe subdivision method is a discrete curve and surface representation based on mesh refinement, it can construct smooth surfaces from any topological mesh. The basic process is that we give a low-resolution control mesh firstly, and define the subdivision rules on the mesh, with the refinement of the mesh we can finally get a smooth surface (limit surface) [1]. Automobile, aircraft, shipbuilding and other manufacturing design, can be subdivided according to the specific subdivision rules, rapid iteration and generate the required model surface.According to whether the limit surface is interpolated or not, the subdivision method can be divided into two types: The first kind is the approximating subdivision schemes, such as quadrilateral mesh-based Catmull-Clark subdivision method [2] and the triangular mesh-based Loop subdivision method [3]. The limit surfaces of the two subdivision methods reach 1C continuous at extraordinary vertex and 2C continuous at the other regular points, and the limit surface shape generated by the approximation subdivision method will shrink with respect to the initial grid. The second kind is the interpolatory subdivision schemes, which subdivides the generated limit surface into a given initial control grid vertex. Such as Dyn [4] presented a butterfly subdivision method, but the method is only applicable to the mesh within the regular points and boundary points, and its subdivision surface at the singular point can only be 0C continuous. Due to lower continuity and restrictions on the initial control mesh, Butterfly subdivision method cannot be applied to any triangle meshes, Denis Zorin presented an modified butterfly subdivision scheme based on the Butterfly subdivision scheme to generate1C continuous surfaces on arbitrary triangular meshes [5]. The limit surface obtained by interpolatory subdivision schemes does not shrink, but the limit surface will produce unnecessary distortion. According to the initial control mesh, the shape of the corresponding limit surface cannot be judged well.There is no analytic expression for the subdivision surface, so the analysis of the surface continuity and smoothness is more difficult. As the subdivision curve control polygon topology is relatively simple, therefore, the convergence and continuity of the univariate subdivision (curve) scheme has been formed into a complete framework [6-8]. For a linear static two-variable subdivision scheme, the continuity analysis is more simply when the control grid is a regular case. The local subdivision matrix can be obtained from the iterative relation, as most of the subdivision matrix has some characteristics of cyclic matrix, therefore, the Discrete Fourier Transform (DFT) can be used to simplify the solution of the eigenvalue of the matrix to reveal the convergence of the iterative process. The remarkable characteristic of the subdivision method is that it cannot be restricted by thecontrol topology mesh and can be used to shape any arbitrary topology mesh. But the existing problem is that the analysis process is complex and unfavorable to the subdivision surface properties[9-10]. Therefore, it is particularly important to find a simple and rapid method to analyze the properties of subdivision surfaces. And interpolation method is the process of finding simple function to approximately replace complex function. This paper presents a Catmull-Clark subdivision method based on the three-dimensional space curve and surface interpolation theory to describe the regular mesh, and builds the mathematical model of Catmull-Clark subdivision method based on curve and surface.2. Catmull-Clark Subdivision Method and Its Limit Point FormulaCatmull and Clark extend the subdivision of bi-cubic B-spline surface into arbitrary topological structure, and their present Catmull-Clark subdivision scheme [2]. Given an initial arbitrary control-points meshes, after a Catmull-Clark subdivision, all control meshes become quadrilateral meshes. For quadrilateral control meshes, the Catmull-Clark subdivision scheme uses 1-4 quadrilateral splits to generate new mesh topologies. The subdivision rule consists of two parts: the generating of geometric point and the creation of topology.2.1 Geometrical Rules(1) New face points: the average of all of the old points defining the face. Set the vertex of the surface of n PP P P ,,,,210 , the new face point, is given by 11nF i i P P n ==∑ (1) On a quadrilateral mesh, is given by12341()4F P P P P P =+++ (2) (2) New edge points: the average of the midpoints of the old edge with the average of the two new face points of the faces sharing the edge. Suppose that i P and j P are the two endpoints of the inner edge, the adjacent new points of the shared inner edge i j PP are 1Fand 2F , and the new edge points is given by 121()4E i j P P PF F =+++ (3) On the quadrilateral control mesh, assuming 01,P P are two ends of an inner edge, 0123P PP P and 0145P PP P are two adjacent quadrilateral faces, and the new edge points is given by01234531()()816E P P P P P P P =+++++ (4) (3) New vertex points: suppose that the vertices around the interior vertex P are 1232,,,,i P P P P , where 2-1i P is the adjacent vertex of the vertex, 2i P is the diagonal vertex on the quadrilateral surface (1,2,3,,)i n = , and the new vertex points is given by22111n n V n n i n i i i P P P P αβγ−===++∑∑ (5) Where 22311(),,24n n n n n n n nαβγβγ=−+==，similar to (5), equation (5) can be converted to 123v n P Q R V n n n−=++ (6) Where Q is the average of the vertices in the face; R is the average of the midpoints of all old edges incident on the old vertex point; P is old vertex point.On the quadrilateral control grid, and the new vertex points is given by111424v P Q R V =++ (7)2.2 Topological RulesAfter these points have been computed, new edges are formed by(1) Connecting each new face point to the new edge points of the edges defining the old face.(2) Connecting each new vertex point to the new edge points of all old edges incident on the old vertex point.2.3 Limit Point FormulaHalstead[11] presented the limit point of the formula. Let (0)v be any vertex on initial mesh (0)M , which corresponds to the point at which vertex ()v ∞ converges to the limit surface in the Catmull-Clark subdivision process. And ()v ∞ is given by2(0)()4(5)j jj j n v e f v n n ∞++=+∑∑ (8) Where n is the degree of the vertex ，(0)j e is the adjacent point in the (0)v1-field, and (0)j f is the non-adjacent point on the (0)v 1-domain surface.3. Mathematical Model of Catmull-Clark Subdivision Scheme on Regular MeshFig. 1 Catmull-Clark subdivision scheme on regular meshIn the regular mesh, the geometrical rules of Catmull-Clark subdivision, new face points, new edge points and new vertex generation follow (2), (3) and (7), and the subdivision is shown in Fig.1. According to Catmull-Clark subdivision rule in the regular quadrilateral mesh, the curve and surface interpolation method is used to calculate the new points. The general form of the resulting limit surface be expressed in matrix form by(,)T T S u v UMPM V = (9) Where ()()32321,1U u u u V v v v == are the parameter vectors, and00102030011121310212223203132333P P P P P P P P P P P P P P P P P = is the set of control points, which are arranged on a topologically quadrilateral mesh according to their subscripts, and0123012301230123a a a a b b b b M c c c c d d d d =is the matrix to be solved.De-Boor algorithm is mainly used to generate continuous curves from discrete points according to algorithm rules, and extend the algorithm to 3D space curves and surfaces, and get generalized De-Boor algorithm to generate space curves and surfaces. In mathematics, the method of generating curves and surfaces conforms to the idea of interpolation, so it can be regarded as the interpolation scheme, according to the curve and surface interpolation theory, combined with subdivision rules, set the parameter values, there,000102101112202122112121211121(0,0)()()()663636366636S P P P P P P P P P =++++++++ (10) 101112202122303132112121211121(0,1)()()()663636366636S P P P P P P P P P =++++++++ (11) 010203111213212223112121211121(1,0)()()()663636366636S P P P P P P P P P =++++++++ (12) 111213212223313233112121211121(1,1)()()()663636366636S P P P P P P P P P =++++++++ (13) 0001021011122021223031321112123121(0,)()()24863648636231211121()()4863648636S P P P P P P P P P P P P =+++++++++++ (14) 0001021011122021223031321412131121(0,)()()38163654636101211121()()27636162636S P P P P P P P P P P P P =+++++++++++ (15) According to the general form of the initial limit surface, bring in the parameter values set in (9), the points are given by0001012023030100111212313102012122232320301312323333000102101112202122(0,0)()()()()112121211121()()()663636366636S d P d P d P d P d d P d P d P d P d d P d P d P d P d d P d P d P d P d P P P P P P P P P +++++++++++++++=++++++++ (16)00010120230300000101112123131111020121222323222203013123233333331011122021(0,1)()()()()()()()()1121212()(6636363S d P d P d P d P a b c d d P d P d P d P a b c d d P d P d P d P a b c d d P d P d P d P a b c d P P P P P =+++++++++++++++++++++++++++=+++++223031321)61121()6636P P P P +++(17)00000010120230301110101112123131222020121222323233303013123233330001021(0,)()()2842()()842()()842()()84211212()48636a b c S d P d P d P d P d a b c d P d P d P d P d a b c d P d P d P d P d a b c d P d P d P d P d P P P =+++++++++++++++++++++++++++=+++1011122021223031323121()48636231211121()()4863648636P P P P P P P P P ++++++++ (18) 0000001012023030111010111212313122202012122232323330301312323333000101(0,)()()32793()()2793()()2793()()27934121(81636a b c S d P d P d P d P d a b c d P d P d P d P d a b c d P d P d P d P d a b c d P d P d P d P d P P P =+++++++++++++++++++++++++++=++210111220212230313231121)()54636101211121()()27636162636P P P P P P P P P +++++++++ (19) For all equations in (16), (17), (18) and (19) to be constant, their corresponding coefficients should be equal, there20000000111121022222220333330000011101222023301()0361()43691()19361()0361()84228823()84228823()842288(8d a b c d d d a b c d d d a b c d d d a b c d d a b c d d a b c d d a b c d d a b d +++= = +++= = +++= = +++= = +++=+++=+++=+00000111012220233333034()279348631()279332410()279316211)()422882793972a b c d d a b c d d a b c d d c a b c d d d +++= +++= +++= ++=+++= (20) Combining the four equations in (20), and solving equations, then there0011112222333311116622121023111122260016adb ca b cdb cadb cd a=−===−==−=====−=====(21)Another set of solutions of the system does not meet the subdivision rules of the Catmull-Clark curve subdivision, so it does not conform to the subdivision rule of the Catmull-Clark subdivision on the surface. Then there012301230123012311116226133111103630122113030600221410121636a a a ab b b bMc c c cd d d d−−−−−−===−−(22)The mathematical expression of the limit surface generated by the Catmull-Clark subdivision method on the regular mesh is given by()300102030201112131320212223203132333(,)1331133136303604130303331141010001 =36T TS u v UMPM VP P P P vP P P P v u u uP P P P vP P P P=−−−−−−−−(23)In the quadrilateral initial control grid of Fig. 1, using the limit point formula (8), the corresponding limit point position on the limit surface is:()11211201100020220211164()()36P P P P P P P P PP∞++++++++=(24)()21201122313032121021164()()36P P P P P P P P PP∞++++++++=(25)()12112213022123130112164()()36P P P P P P P P PP∞++++++++=(26)()22211223323133111322164()()36P P P P P P P P PP∞++++++++=(27) After simplification (10), (11), (12), (13) ,with the formula (24), (25), (26) , (27) comparison, then there()()()()11211222(0,0)(0,1)(1,0)(1,1)S P S P S P S P∞∞∞∞====，，，(28) The subdivision model of the Catmull-Clark subdivision method in the regular mesh is established, and the resulting limit surface is a bi-cubic B-spline surface. From the established mathematical model, it is obvious that the properties of the convergence, differentiability and curvature of the subdivision surface can be directly studied.4. ConclusionsAccording to the subdivision rule of the Catmull-Clark subdivision method in the regular mesh we describe the subdivision rule by the 3D space surface interpolation theory, then the mathematic expression of the limit surface is obtained by using undetermined coefficient method, and the mathematic model of the subdivision method is established. The results show that the limit surface generated by the Catmull-Clark subdivision method in the regular mesh is a tensor product bi-cubic B-spline surface [12].Using mathematical analysis, the mathematic model established above can be used to study the properties of subdivision surfaces, such as convergence, continuity, differentiability, and so on, and the guide vector and curvature of the surface also can be obtained. Using the discrete Fourier transform to analyze the properties of limit surfaces is more simply than using local subdivision matrices.In this paper, only the mathematical model of Catmull-Clark subdivision method in the regular mesh is established, and the initial control mesh is not considered. So we can focus on the mathematical model of limit surface at irregular mesh.5. References[1] Schröder P, Zorin D, DeRose T, et al. Subdivision for modeling and animation[J]. ACM SIGGRAPH Course Notes, 1998, 12.[2] Catmull E, Clark J. Recursively generated B-spline surfaces on arbitrary topological meshes[J]. Computer-aided design, 1978, 10(6): 350-355.[3] Loop C. Smooth subdivision surfaces based on triangles[D]. [Master’s thesis]. Salt Lake City: Department of Mathematics University of Utah, 1987.[4] Dyn N, Levine D, Gregory J A. A butterfly subdivision scheme for surface interpolation with tension control[J]. ACM transactions on Graphics (TOG), 1990, 9(2): 160-169.[5] Zorin D, Schröder P, Sweldens W. Interpolating subdivision for meshes with arbitrary topology[C]//Proceedings of the 23rd annual conference on Computer graphics and interactive techniques. ACM, 1996: 189-192[6] Cavaretta A S, Dahmen W, Micchelli C A. Stationary subdivision[M]. American Mathematical Soc., 1991 , 93(453):186.[7] Micchelli C A. Mathematical Aspects of Geometric Modeling[M]// Mathematical aspects of geometric modeling. Society for Industrial and Applied Mathematics, 1995.[8] Warren J, Weimer H. Subdivision methods for geometric design: A constructive approach[M]. Morgan Kaufmann, 2001:276-285.[9] Stam J. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values[C]//Proceedings of the 25th annual conference on Computer graphics and interactive techniques. ACM, 1998: 395-404.[10] Stam J. Evaluation of Loop subdivision surfaces [C]//Proceedings of the 25th annual conference on Computer graphics and interactive techniques. ACM, 1998: 405-417.[11] Halstead M, Kass M, DeRose T. Efficient, fair interpolation using Catmull-Clark surfaces[C]//Proceedings of the 20th annual conference on Computer graphics and interactive techniques. ACM, 1993: 35-44.[12] CHEN Wei, WANG Rong-Qing, et al. Weighted Catmull-Clark Surface[J]. Journal of Computer- Aided Design & Computer Graphics 2004, 12(4): 277-280.。

Connection Elements and ConnectionLibraryLecture 2Copyright 2006 ABAQUS, Inc.L2.2 Overview•Introduction•Defining Connector Elements•Understanding Connector Sections•Understanding Connection Types•Understanding Connector Local Directions•Rotational Degrees of Freedom at Nodes•Components of Relative Motion•Connector Local Kinematics•Summary of Orientations and Local DirectionsFlexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Copyright 2006 ABAQUS, Inc.IntroductionFlexible Multibody Systems with ABAQUSL2.4Copyright 2006 ABAQUS, Inc.Introduction•General characteristics of connector elements•Connector elements model discrete (point-to-point) physical connections between deformable or rigid bodies or can be connected to ground.•Example: typical connections in automotive suspension systemsTypical connections in automotive suspensionRack and pinion Control armTie rodKnuckleStrutJOIN CYLINDRICALAXIALCYLINDRICALLINKFlexible Multibody Systems with ABAQUSL2.5Copyright 2006 ABAQUS, Inc.Introduction•Connector elements have relative displacements and rotations that are local to the element, which are referred to as components of relative motions (CORM).•Connector elements impose kinematic constraints. For example:•Door connected to a frame through a hinge •Two panels of sheet metal spot welded together •Constant velocity joints•Connector elements may include (nonlinear) force-versus-displacement (or velocity) behavior in their available components of relative motion. •Example: muscle force resisting the rotation of a knee joint in a crash-test occupant dummy model.•Connector elements can provide comprehensive kinematic and kinetic output describing the connection.Flexible Multibody Systems with ABAQUSL2.6Copyright 2006 ABAQUS, Inc.Introduction•Connector elements are functionally defined by specifying the connection attributes.•Connection types•For example: axial, hinge, weld, constant velocity joints, link,beam, etc.•Local connector directions•Connector behaviors•Uncoupled or coupled response•Linear or nonlinear response •Elasticity and damping •Plasticity •Friction•Damage and Failure •Stops and locks AXIAL connection in a shock absorbingstrut uses a variety of connection behaviors, including spring, damping and stopbehaviorsAXIALDefining Connector ElementsCopyright 2006 ABAQUS, Inc.L2.8 Defining Connector Elements•Connector elements are 2-node elements.•Element type:•CONN2D2•Two-dimensional analysis•Axisymmetric analysis•CONN3D2•Three-dimensional analysis•Both types of element have at most two nodes.•The position (location, orientation) and motion (displacement, velocity,acceleration) of the second node on the element are measured relative tothe first node.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.9Copyright 2006 ABAQUS, Inc.Defining Connector Elements•Defining a connector element –Keywords interface•To connect two points:*ELEMENT, TYPE=[CONN2D2 or CONN3D2]element number , first node number , second node number•Example: Shock absorber*ELEMENT, TYPE=CONN3D2101, 11, 12Simplified connector model of a shock absorberab1Flexible Multibody Systems with ABAQUSL2.10Copyright 2006 ABAQUS, Inc.Defining Connector Elements•To connect a point to ground:•The ground “node”can be the first or second node on the connectorelement.*ELEMENT, TYPE=nameelement number , , node number on the bodyor*ELEMENT, TYPE=nameelement number , node number on the body•The ground node is fixed.2L2.11 Defining Connector Elements•Defining connector geometry –ABAQUS/CAE interface•Create assembly-level wire features to defineconnector geometry in the Interaction module.•disjoint wires•chained wires•wires to ground.•Click Add to add points•Tip: two coincident points can beselected simultaneously by double-clickingon the location of the points•Click Delete to delete the selected point pair.•Click Swap to swap the points of the selectedpoint pair.•Tip: the point pair needs to be selected byclick its index number.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.12 Defining Connector Elements• A geometry set including all of the wires can becreated when creating the wire feature.•The set can be used during the subsequentselection procedures.•For example, you can use sets to assignconnector sections, request output, orprescribe motions.•Note: Multiple sets can be mergedinto a new set using the set mergefeature.•Assembly-level wire features cannot be modifieddirectly once created.•Wires can be removed by selecting RemoveWires From Feature.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.13Copyright 2006 ABAQUS, Inc.Defining Connector Elements•Example: Truck door hinges•In this example, hinge connectors connect a truck door to the truck body.ABAQUS/CAE interface Create wire geometryKeywords interface Define a connector element*ELEMENT, TYPE=CONN3D2, ELSET=CONN_DOOR_HINGE 620601, 9000009, 9000010620602, 9000011, 9000012...2 nodes per elementRP-2node 9000010HINGE connectorsat door hingesCopyright 2006 ABAQUS, Inc.Understanding Connector SectionsFlexible Multibody Systems with ABAQUSL2.15Copyright 2006 ABAQUS, Inc.Understanding Connector Sections•Connector section defines:•The connection type.•The local directions associated with the connector’s nodes •The connector behaviors.•Note: Details of connector behavior will be discussed in lectures 4 and 5.•Creating a connector section*CONNECTOR SECTION, ELSET=nameABAQUS/CAE interfaceKeywords interfaceFlexible Multibody Systems with ABAQUSL2.16Copyright 2006 ABAQUS, Inc.Understanding Connector Sections•Defining the connection type•Basic connection components•Translations •Rotations•Assembled connection components•Combination of basic connection componentsKeywords interfaceABAQUS/CAE interface*CONNECTOR SECTION, ELSET =name basic connection type, <basic connection type>or*CONNECTOR SECTION, ELSET=name assembled connectionFlexible Multibody Systems with ABAQUSL2.17Copyright 2006 ABAQUS, Inc.Understanding Connector Sections•Example: Truck door hinges –Keywords interface•Define and assign a connector section*ELEMENT, TYPE=CONN3D2, ELSET=CONN_DOOR_HINGE 620601,9000009,9000010620602,9000011,9000012*CONNECTOR SECTION , ELSET=CONN_DOOR_HINGE HINGE ...Assembled connection; one can also use basic connection types: JOIN and REVOLUTE .Connector elements CONN_DOOR_HINGE(assignment)Flexible Multibody Systems with ABAQUSL2.18Copyright 2006 ABAQUS, Inc.Understanding Connector Sections•Example: Truck door hinges –ABAQUS/CAE interface•Define and assign a connector section•Define a connector sectionAssembled Hinge connection;one can also use basic connection types: JOIN and REVOLUTE.No behavior options are specified (default).Note: Details will be discussed in lectures 4 and 5.1Flexible Multibody Systems with ABAQUSL2.19Copyright 2006 ABAQUS, Inc.Understanding Connector Sections•Assign a connector section•Connector section assignment is used to assign a connector section to a region (wires) of the model.Assign the connection section DOOR_HINGES to the wire Wire-1-DOOR_HINGES .2Copyright 2006 ABAQUS, Inc.Understanding Connection TypesFlexible Multibody Systems with ABAQUSL2.21Copyright 2006 ABAQUS, Inc.Understanding Connection Types•Connection types•The connection-type library contains:•Translational basic connection components, which affecttranslational DOFs at both nodes and may affect rotational DOFs at the first node of the connector element.•Rotational basic connection components, which affect only rotational DOFs at both nodes of the connector element •Assembled connections , which are a predefined combination of translational and rotational basic connection components.•The above choices determine which element local DOFs exist.•Given the number of connection types available, it is clear that connector elements can easily be customized to suit an application.Flexible Multibody Systems with ABAQUSL2.22Copyright 2006 ABAQUS, Inc.•Examples of translational basic connections :•AXIAL –Provide a connection between two nodes that acts along the line connecting the nodes.•CARTESIAN –Provide a connection between two nodes that allows independent behavior in three local Cartesian directions.•JOIN –Join the position of two nodes.•ACCELEROMETER –Provide a connection between two nodes to measure the relative acceleration, velocity, and position of a body in a local coordinate system.•Available only in 3D analysis in ABAQUS/Explicit. •Will be converted internally to a CARTESIAN connector type in ABAQUS/Standard.bau 1b ab aAXIALJOINCARTESIANACCELEROMETERUnderstanding Connection TypesFlexible Multibody Systems with ABAQUSL2.23Copyright 2006 ABAQUS, Inc.•Examples of rotational basic connections:•REVOLUTE –Provides a revolute connection between two nodes.•CARDAN –Provides arotational connection between two nodes parameterized by Cardan angles.•EULER –Provides a rotational connection between two nodes parameterized by Euler anglesREVOLUTECARDANEULERUnderstanding Connection TypesFlexible Multibody Systems with ABAQUSL2.24Copyright 2006 ABAQUS, Inc.Understanding Connection TypesBasic rotational Basic translational ROTATION-ACCELEROMETER SLOTROTATIONSLIDE-PLANE UNIVERSALREVOLUTE RADIAL-THRUST PROJECTION FLEXION-TORSION PROJECTION CARTESIAN FLEXION-TORSIONLINKEULERJOIN CONSTANT VELOCITY CARTESIAN CARDANAXIAL ALIGN ACCELEROMETER ab aSummary of basic connection typesFlexible Multibody Systems with ABAQUSL2.25Copyright 2006 ABAQUS, Inc.•Examples of assembled connections:•BEAM –provides a rigid beam connection between two nodes (JOIN + ALIGN)•HINGE –joins the position of two nodes, and provides a revolute connection between their rotational degrees of freedom (JOIN + REVOLUTE)•UJOINT –joins the position of two nodes, and provides a universal connection between their rotational degrees of freedom at the nodes (JOIN + UNIVERSAL)abBEAMHINGE UJOINTUnderstanding Connection TypesFlexible Multibody Systems with ABAQUSL2.26Copyright 2006 ABAQUS, Inc.Understanding Connection TypesJOINJOIN SLOT SLIDE-PLANE JOINSLOT JOIN PROJECTION CARTESIAN JOINEquivalent basic connection components(translational + rotational)Assembled ALIGNWELDUNIVERSAL UJOINT ALIGN TRANSLATOR REVOLUTE PLANAR REVOLUTE HINGE REVOLUTE CYLINDRICAL CONSTANT VELOCITY CVJOINT PROJECTIONFLEXION-TORSION BUSHINGALIGNBEAMSummary of assembled connection typesL2.27 Understanding Connection Types•More advanced connection types, such as FLOW-CONVERTER,RETRACTOR and SLIPRING, are also available.•These connectors are special pulley-type connectors used to modelseatbelt kinematics.•These connection types will not be discussed here but are discussedfurther in Appendix1.•Example: Three-point belt model with retractor and pretensioner.Video ClipVideo Clip Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Understanding Connector LocalDirectionsCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.29Copyright 2006 ABAQUS, Inc.•Understanding connector local directions•Orientations are used to define local directions for connection types that use local directions (local directions may be required or optional)•The local directions are defined by reference to a local orientation or coordinate system.•Example: HINGE connector requires an orientation to be associated with the first node a .•The hinge axis is aligned with the orientation X -direction.(local orientation)XUnderstanding Connector Local DirectionsFlexible Multibody Systems with ABAQUSL2.30Copyright 2006 ABAQUS, Inc.•Not all connection type require the local directions (e.g. LINK).•In some cases default local directions are chosen (e.g. CARTESIAN)•Local directions at the second node are not used by all connection types.•Example: SLOT connection•The line of the slot is defined by the first local direction at node a and the initial position of node b . (fig. a)•The SLOT connection constrains the position of node b (x b ) to remain on the line of the slot.•Note: different results would be obtained if different orientations are used for the local directions. (figs. b, c)21u 1u 112b. Orientation pointing from node a to node bc. 45o counterclockwise rotation of the slota. SLOT connectionaUnderstanding Connector Local DirectionsFlexible Multibody Systems with ABAQUSL2.31Copyright 2006 ABAQUS, Inc.Understanding Connector Local Directions•The default directions at the second node are the local directions at the first node.•It may be necessary to define local directions at the second node to model the mechanism correctly, e.g. UJOINT.•In geometrically nonlinear analyses, the element local directionsassociated with the nodes rotate with the rotational degrees of freedom at the nodes.• A summary of connector local directions will be in the section “Summary of Orientations and Local Directions”in this lecture.Flexible Multibody Systems with ABAQUSL2.32Copyright 2006 ABAQUS, Inc.•Defining connector orientation•Example: Truck door hingesZ YXORI_CONN_DOORABAQUS/CAE interface Define orientation using a datum coordinate systemKeywords interfaceDefine orientation using local CSYS*ORIENTATION , NAME=ORI_CONN_DOOR 0.,0.,1., 1.,0.,0.3,0Understanding Connector Local DirectionsFlexible Multibody Systems with ABAQUSL2.33Copyright 2006 ABAQUS, Inc.•Example: Truck door hinges*ORIENTATION, NAME=ORI_CONN_DOOR 0.,0.,1.,1.,0.,0.3,0*ELEMENT, TYPE=CONN3D2, ELSET=CONN_DOOR_HINGE 620601,9000009,9000010620602,9000011,9000012*CONNECTOR SECTION, ELSET=CONN_DOOR_HINGE HINGEORI_CONN_DOORConnector elements CONN_DOOR_HINGEKeywords interfaceZ YXORI_CONN_DOORUnderstanding Connector Local DirectionsFlexible Multibody Systems with ABAQUSL2.34Copyright 2006 ABAQUS, Inc.•Specify local orientations for the endpoints of the wires –ABAQUS/CAE interface.•Connector section assignment is used to specify local orientations for the endpoints of the wires.•Example: Truck door hingesZ YXORI_CONN_DOORUnderstanding Connector Local DirectionsRotational Degrees of Freedom at theNodesCopyright 2006 ABAQUS, Inc.L2.36 Rotational Degrees of Freedom at the Nodes•Overview•In geometrically nonlinear analyses, the element local directionsassociated with the nodes rotate with the rotational degrees offreedom at the nodes.•In linear or perturbation analyses, the element local directionsremain fixed.•In cases where an orientation definition is permitted for definingconnection directions (either required or optional), the connectorelement will activate rotational degrees of freedom at the nodes if theydo not exist already.•The only exception is JOIN (will be discussed later).Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.37 Rotational Degrees of Freedom at the Nodes•Example: The BEAM connection activates rotational degrees of freedom.The solid elements do not provide rotational stiffness at these DOFs; theconnector does NOT transmit rotation into the solid.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.38 Rotational Degrees of Freedom at the Nodes•Other connections where an orientation definition is permitted activaterotational degrees of freedom (e.g. BEAM, …):•If local directions are used and either the element's nodes do not possessrotational DOFs or if rotational constraints such as,•Equations•Multi-point constraints•Boundary conditionsare not applied to the nodes, numerical singularities associated withunconstrained degrees of freedom will exist.•Solutions:•Attach the connector element to a structural element.•Add rotational boundary conditions.•Use coupling constraints (Recommended)Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.39Copyright 2006 ABAQUS, Inc.Rotational Degrees of Freedom at the Nodes•JOIN connection does NOT activate rotational degrees of freedom•This allows the user to define a join constraint between two solids expressed only in terms of translations.•Example 1: Two deformable solids connected by the JOIN connection type rotate under certain loading and boundary conditions.•Orientation at node a does not rotate with the rotation of thedeformable body since the solid element does not have rotational DOFs actived.Deformable12XY ZJOINDeformable C3D8RC3D8RDeformable CLOADCLOADDeformable12JOINabNote: The examples discussed here and the next two slides consider geometric nonlinearity.Flexible Multibody Systems with ABAQUSL2.40Copyright 2006 ABAQUS, Inc.Rotational Degrees of Freedom at the Nodes•Example 2: Make one solid in Example 1rigid.•Orientation at node a rotates with the rotation of the rigid body since the rigid body has rotational DOFs.•Node b will move accordingly with node a .12XY ZJOINRigid C3D8RC3D8RDeformable CLOADCLOAD12J O I N RigidDeformableRPa bFlexible Multibody Systems with ABAQUSL2.41Copyright 2006 ABAQUS, Inc.Rotational Degrees of Freedom at the NodesXY ZJOIN b213aSurface-basedcoupling constraintDeformableC3D8R DeformableC3D8R12J O I N DeformableDeformableCLOADCLOAD•Example 3: Define a surface-based coupling constraint on the surface of one of the deformable solids in Example 1; choose the reference point of the surface-based coupling constraint as node a .•Orientation at node a rotates with the rotation of the deformable body since the coupling constraint actives rotational DOFs.•Node b will move accordingly with node a .Copyright 2006 ABAQUS, Inc.Components of Relative MotionL2.43 Components of Relative Motion•Components of relative motion•Connector elements have internal DOFs that do not exist at any node,but are a part of the connector element itself.•The connector local degrees of freedom, that is, the three translationsand three rotations relative to the connector element local coordinatesystem (in three dimensions), are called the components of relativemotion (CORM).•The three translations are in the element local coordinate directions.•The three rotations are angular quantities that depend on thespecific connection definition and may or may not be rotations aboutorthogonal directions.•All components of relative motion are either constrained or available.•The definitions of constrained and available components of relativemotion will be discussed in next two slides, respectively.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.44 Components of Relative Motion•Constrained components of relative motion•Constrained components of relative motion are displacements androtations that are fixed by the connector element.•ABAQUS/Standard uses Lagrange multipliers to enforce the kinematicconstraints.•The constraint forces and moments carried by the element appearas additional solution variables.•ABAQUS/Explicit uses an augmented Lagrange technique to enforce thekinematic constraints.•Constrained components of relative motion are measured by reactionforces and moments.•The kinematic constraint is equivalent to forcing one or more of thecomponents of relative motion to behave according to some predefinedrelationship.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.45Copyright 2006 ABAQUS, Inc.Components of Relative Motion•Available components of relative motion•Available components of relative motion are displacements and rotations that are not constrained kinematically.•Available components of relative motion can be used for•Defining material-like behavior (discussed in lectures 4 and 5).•Specifying connector actuation (discussed in lecture 6).•Time-dependent motion •Applying loading•Assigning complex interactions, such as contact or friction (discussed in lecture 4).Flexible Multibody Systems with ABAQUSL2.46Copyright 2006 ABAQUS, Inc.Components of Relative Motion•Examples:3+1:3+2:3+3:0:2: 3: u 1=u 2=u 3=00:0:Kinematic constraints 2: ur 1, ur 3U-JOINT1: ur 1HINGE 0:BEAM 3: ur 1, ur 2, ur 3CARDAN/EULER 1: ur 1REVOLUTE 0:JOIN 3: u 1, u 2, u 3CARTESIAN 1: u 1AXIAL Available components Connection types ⋅=⋅=a b a b 12130,0e e e e ⋅=⋅=a b a b 12130,0e e e e u 1=u 2=u 3=0u 1=u 2=u 3=0⋅=a b 130e e ur 1=ur 2=ur 3=0u 1=u 2=u 3=0–For other connection types refer to “Connection-type library”section 25.1.5 in ABAQUS Analysis User’sManual.Connector Local KinematicsCopyright 2006 ABAQUS, Inc.L2.48 Connector Local Kinematics•There are many different kinematic quantities used to describe aconnector element:•Position•Displacement•Velocity•Acceleration•Reference position for defining constitutive response•Constitutive displacements (essentially material strains)•All these quantities provide useful information regarding the connectorelement.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.49 Connector Local Kinematics•In three dimensions each of the above kinematic quantities requires sixcomponents to completely define it.•Three components in two dimensions.•For example, in three dimensions, a position requires•Three generalized coordinates to identify the location of the secondnode relative to the first.•Three angles to identify the orientation of the second node relativeto the first.•In two dimensions, a position requires•Two generalized coordinates to identify the location of the secondnode relative to the first.•One angle to identify the orientation of the second node relative tothe first.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.50 Connector Local Kinematics•Displacement, velocity, acceleration•These quantities are derived from position:•Displacement = change in position•Velocity = rate of change of position•Acceleration = rate of change of velocity•The reference position for constitutive response and constitutivedisplacements is relevant only for material behavior and will bediscussed in Lecture 4.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.51 Connector Local Kinematics•Translations:•With the exception of RADIAL-THRUST, BUSHING, PROJECTIONCARTESIAN and AXIAL, all the connections use Cartesian coordinatesto locate the second node relative to the first.•For translations the second node’s position, displacement, velocity,etc. are identified by three Cartesian components relative to localdirections at the connector element’s first node.•RADIAL-THRUST uses a cylindrical coordinate system with origin at thefirst node.•BUSHING and PROJECTION CARTESIAN use an orthonormal systemthat follows the systems at both nodes in the connection.•AXIAL does not use a coordinate system. The CORM is measured alongthe line separating the two nodes in the connection.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.52 Connector Local Kinematics•Rotations:•Describing directions at the second node relative to directions at the firstnode is much more complicated.•Example: REVOLUTE•Several different rotation parameterizations are used.•Cardan angles and Euler angles are successive rotation parameterizations.Examples:•Spinning top = EULER angles (precession, nutation, and spin).•Airplane attitude = CARDAN angles (roll, pitch, and yaw).Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.L2.53 Connector Local Kinematics•The rotation vector is a parameterization similar to the nodal rotationaldegrees of freedom in ABAQUS.•Example: Instantaneous angular velocity•FLEXION-TORSION has rotation parameterization angles consisting of totalflexion, torsion, and sweep.•PROJECTION FLEXION-TORSION connection has rotationparameterization angles consisting of two component flexion angles and atorsion angle.•Example: Head position on the shoulders in a crash test dummy model.•Different rotation parameterizations are analogous to different coordinate systems (Cartesian versus cylindrical versus polar, etc.).•Use the one that makes the most sense for a particular application.Flexible Multibody Systems with ABAQUSCopyright 2006 ABAQUS, Inc.Summary of Orientation and LocalDirectionsCopyright 2006 ABAQUS, Inc.Flexible Multibody Systems with ABAQUSL2.55Copyright 2006 ABAQUS, Inc.•Summary of connector local directions in different connection types.Summary of Orientation and Local DirectionsOptional **Optional **AXIAL Ignored Required RADIAL-THRUST Ignored Ignored LINKIgnored Optional *JOIN Optional Optional ACCELEROMETEROptional Optional PROJECTION CARTESIANIgnoredRequiredSLOTIgnored Required SLIDE-PLANEIgnored Optional CARTESIAN2nd node 1st node Connection Type Local Directions at Connector Nodes –Basic Translational*Rotational dofs are not activated. If rotational dofs do not exist, the local coordinate system does not co-rotate and fixed directions are used.**Optional only at a ground node. Otherwise, ignored.Flexible Multibody Systems with ABAQUSL2.56Copyright 2006 ABAQUS, Inc.Summary of Orientation and Local DirectionsLocal Directions at Connector Nodes –Basic RotationalOptional Required UNIVERSAL Optional Optional ROTATION-ACCELEROMETERIgnoredRequiredFLOW-CONVERTEROptional Required REVOLUTEOptional Required PROJECTION FLEXION-TORSIONOptional Required FLEXION-TORSIONOptional Required EULER Optional Required CONSTANT VELOCITYOptional Required CARDANOptional Optional ALIGN Optional Optional ROTATION 2nd node 1st node Connection TypeFlexible Multibody Systems with ABAQUSL2.57Copyright 2006 ABAQUS, Inc.Summary of Orientation and Local DirectionsLocal Directions at Connector Nodes –AssembledOptional Optional WELD IgnoredIgnoredSLIPRINGOptional Required UJOINT Optional Required TRANSLATOROptional Required PLANAR Optional Required HINGE Optional Required CYLINDRICALOptional Required CVJOINT Optional Required BUSING Optional Optional BEAM 2nd node 1st node Connection TypeFlexible Multibody Systems with ABAQUSL2.58Copyright 2006 ABAQUS, Inc.Summary of Orientation and Local Directions•Example: the definitions of orientation and local directions in UJOINTUnit angular velocity at node 1 about shaft 1axis, what is shaft 2rotation after 1 sec?*ELEMENT, TYPE=CONN3D2, ELSET=UJOINT_EL3,2,3*CONNECTOR SECTION, ELSET=UJOINT_ELUJOINT,ORI_SHAFT_1,ORI_SHAFT_2*ORIENTATION, NAME=ORI_SHAFT_10.0,0.0,1.0, 1.0,0.0,0.0*ORIENTATION, NAME=ORI_SHAFT_20.0,0.0,1.0, 0.937,0.342,0.0*TRANSFORM, NSET=NODE40. 937,0.342,0.0, -0.342,0.937,0.0*BOUNDARY, TYPE=VELOCITY 1,4,4,1.0*NODE PRINT, NSET=NODE4, GLOBAL=NO U,β=20oω1ω2Shaft 1Shaft 21234UJOINT:if drive shaft moves at constant angular velocity, for β≠0, driven shaft moves with uneven angular velocity (it “wobbles”)driven shaft twist angle = α2= arctan[cos(β) tan(ω1t )]local 2 axes along shaft axesfix U 1,U 2,U 3fix UR 2,UR 3fix U 1,U 2,U 3UR 1 in local coord. system is 0.971(Note: arctan[cos(20ο) tan(1.0∗1.0)]=0.971O.K.)align local 1 direction at node 4 with shaft 2axisStep time = 1.0sec1global axes231’2’3’1’2’3’。

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It helps teachers to support students to achieve their potential and seamlessly progress to the next level of learning, through the effective integration of print and digital content and activities.Why bilArabi?••in children.• and techniques in the classrooms.••Our standardsOur contentLiterature-basedA mix of photos and illustrations, plus interactive videos engages students in ArabicMore than just an Arabic language learning programmeWith a comprehensive range of informational and literature reading texts, children are fully immersed in reading and writing from the beginning of the course, helping them to become better readers, writers and thinkers. Engaging stories completely engage students in the Arabic language, accelerating language proficiency and progress.bilArabi has a vibrant look and feel that uses a blend of illustrations and real-life photos to engage students with the course content. 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