Geometrically Exact Beam Formulation versus Absolute Nodal Coordinate Formulation

Transcription

1 he 1 st Joint International Conference on Multibody Syste Dynaics May appeenranta Finland Geoetrically Exact Bea Forulation versus Absolute Nodal Coordinate Forulation Jari M.A. Mäkinen * Marko K. Matikainen # * Mechanics and Design apere University of echnology P.O. Box apere Finland e-ail: jari..akinen@tut.fi # Mechanical Engineering appeenranta University of echnology Skinnarilankatu appeenranta Finland e-ail: arko.atikainen@lut.fi ABSAC he purpose of this study is to ake coparisons between the geoetrically exact bea forulation (GEBF) and the absolute nodal coordinate forulation (ANCF). In earlier studies the GEBF and ANCF bea forulations are copared in two and three diensional cases [39]. However the dynaics of spatial beas undergoing very large rotations are not involved in the nuerical exaples in [9]. In order to clarify the coputational efficiency of the GEBF and ANCF spatial bea eleents in the case of large rotations ultibody probles singularity-free GEBF ipleentation is used for coparisons in this study. Keywords: Geoetrically exact bea forulation Absolute nodal coordinate forulation 1 INODUCION A bea theory (or odel) is called exact if no other kineatic siplifications during derivation than the basic kineatic assuption like ioshenko eissner bea hypothesis are exploited. In the geoetrically exact bea theory with ioshenko eissner bea hypothesis transversal shear deforations are accounted for while a cross-section reains in-plane and inextensible. Due to the description of shear deforation the bea cross-section is not necessarily parallel with the tangent of the central line. he geoetrically exact bea eleents have been exained in nuerous studies [81146]. his forulation is suitable for ultibody applications in which large deforations i.e. large displaceents and rotations and possibly large strains need to be accounted for. When the theory is applied to practical applications the eleent can be described within the concept of the total agrangian forulation. However to overcoe the singularity proble associated the rotation vector with the rotation angle and ultiplies in the total agrangian forulation the copleent rotation vector is introduced [6]. With paraeterizations for the rotation and the copleent rotation vectors any rotation without singularities can be presented. he finite eleents based on GEBF are discretized using position and rotational nodal coordinates. his discretization leads to a constant description of the ass atrix for two-diensional eleents. However in three-diensional cases discretization leads the ass atrix to no longer be constant regardless of the choice of rotational coordinates. he ANCF is a nonlinear finite eleent approach that is based on the use of global position and gradient coordinates. he forulation is designed for large deforation analysis of bea plate and shell type structures in ultibody applications [11]. he kineatics description of an eleent based on the forulation does not include the rotational degrees of freedo. In this forulation gradient vector coordinates that are partial derivatives of the position vector are used to describe the cross-section orientations and deforations. herefore all nodal coordinates are described in an inertial frae allowing for the usage of the total agrangian approach such as in the case of geoetrically exact bea forulation. he use of this set of generalized coordinates leads to a singularity-free description in large rotation probles and also to a constant ass atrix. his siplifies the description of the equations of otion. Additionally the aterial odels known fro general continuu echanics can be ipleented

2 in the forulation. herefore ANCF eleents can be considered as ore general than classical bea eleents. In this study the GEBF eleent is ipleented within the fraework of the total agrangian forulation without singularity probles. Singularities of the rotation vector are avoided by varying paraeterization on the rotation anifold [6]. Furtherore the use of such paraetrization leads to correct representation of orientation within an eleent. Used ipleentations of ANCF fullyparaeterized eleent are based on the elastic line approach [] and the general continuu echanics. he ain objective of this study is to clarify the coputational efficiency of the GEBF and ANCF eleents in the case of large rotation static and dynaic exaples. In the study the siple three diensional nuerical exaples are solved by using the GEBF and ANCF spatial bea eleents to deonstrate the advantages and disadvantages of eleents. GEOMEICAY EXAC BEAM FOMUAION he virtual work states for a aterial body W x bdv x da F:PdV x x dv 0 (1) B0 B0 B0 B0 where b 0 are the body force vector the density of the aterial body B 0 and the given traction vector on the boundary of the aterial body see detail in [7]. he first two ters of the right-hand side of Eqn (1) correspond to the external virtual work Wext the third ter is the internal virtual work Wint and the last ter is the accelerational virtual work Wacc. he deforation gradient is defined by F: x/ X. he ie derivative of deforation gradient F is equal to F F. he first Piola- Kirchhoff stress tensor is defined by P: i E i where the stress vector i acts on the faces of the defored eleent of volue. et { x1 x x3 } be co-ordinates of a spatially fixed frae and let { X1 X X3 } be co-ordinates of a aterial frae then the aterial point X in the defored placeent field can be given with the aid of the spatial frae { O t t t t } 1 3 as 0 () where s is the bea length paraeter and x c deterines the central line of cross-section. he spatial frae { O t t1 t t3 } coincides with the aterial frae at an initial placeent i.e. ti Ei i 13. Due to shear affects the tangent of central line d d s does not necessarily coincide with the cross-section x c noral vector t 1. A linear interpolation can be applied for the placeent x c and for the spatial base vectors t() s () s E and t 3 () s () s E 3 where the rotation atrix is interpolated via the total rotation vector () s. If the displaceent and rotation fields are interpolated linearly the geoetrically exact bea eleent has totally 1 degrees of freedo (DOFs). educed one-point Gaussian quadrature is used for eliinating the shear locking phenoena. he total rotation vector and the rotation atrix are related by sin 1cos : I exp( ) (3) where is the skew-syetric tensor of its axial vector. he rotation vector () s is a vector quantity and it can be correctly interpolated like translational displaceent see [67]. he virtual work of acceleration forces Wacc reads with aid of the central line conditions W x ( A x )d s ( J J)d s (4) acc c 0 c 0

3 where x c is the bea central line placeent is the virtual increental aterial rotation vector is the aterial angular velocity vector and is the aterial angular acceleration vector see [6]. If the principal axes of the inertial tensor J are parallel to the base vectors { E E 3} then the atrix of the inertial tensor is diagonal. In following this assuption is used for siplicity having J diag( J1 J J3) J1 J J3 where the lower indexes correspond to the base vectors { E1 E E 3}. he virtual work of external forces can be written via the bea kineatic assuptions () Wext xcnds M ds (5) where the external force vector n and the external aterial oent vector M. Next the virtual work of internal forces is derived the third ter in Eqn (1). he deforation gradient F states after substituting the bea kineatic assuptions () where the aterial curvature tensor c 1 1 F x( s) Et E + (6) :. he definition for ie variation reads generating an objective variation. he ie variation of the deforation gradient is c c 1. : F x x E E (7) Now the ter F:P (the virtual work of internal forces) in (1) reads W ( x x) Nds M d s (8) int c c where corresponds to the initial length of the bea. he aterial internal force vector N and the aterial internal oent vector M are denoted by the forulas : 1d A : 1d A. A A N M E (9) where 1 is the stress vector acting on the cross-section face. he work conjugate of the aterial vector N is the variation of the aterial strain vector denoted by (10) : xc E1 xc x c Finally the internal virtual work reads in the aterial representation where the aterial curvature tensor is defined by int W N M ds (11) and where the aterial internal force : vector N and the aterial internal oent vector M are denoted by the constitutive forulas. A linear constitutive relation is introduced in the aterial representation between aterial strain and curvature vectors and internal force and oent vectors by writing N Cn M C Next the virtual work for is derived in ters of the total rotation vector and in order to generate a total agrangian forulation. he spin vectors like the virtual increental rotation vector (1)

4 the angular velocity vector the angular acceleration vector and the curvature vector are required to express in ters of the total rotation vector and giving A (13) where the tangential transforation is given in [6]. It should be noted that at I whereas the spin vectors at. he tangential vector space at is different for distinct rotation atrices. Now the virtual work of acceleration forces Wacc vectors (4) can be written in ters of the total aterial yielding ~ acc xc 0xc J J J W ( A )d s ( ) ds (14) that can be decoposed into the acceleration dependent part WaccA and the velocity dependent part W accb W A s s W acca xc ( 0xc )d J d ~ accb J ( ) J d. s (15) he virtual work of external forces reads in at I -forulation W s s (16) ext xc nd M d where the external aterial oent vector M at so it is a spin vector. According to the above forula a new external aterial oent vector if defined by setting M M (17) I: ati where at I is the initial vector space of rotation in the aterial description. Finally the virtual work of internal forces (11) can be written in the total agrangian forulation ~ xn x N C M M W ( )d s ( ) ( ) ds (18) int c c 1 where tensor C 1 is given [6]. It is clear that the virtual work of internal forces is ore coupled than the other virtual work fors. his is ainly because of the choice of dependent variable the rotation vector. 3 ABSOUE NODA COODINAE FOMUAION In eleents based on the absolute nodal coordinate forulation kineatics can be expressed using spatial shape functions and global coordinates. he kineatics of the fully-paraeterized bea eleent can be written as: where r= S ( x) e= S ( (x)e ) (19) S is a shape function atrix e= e () t is the vector of nodal coordinates and vector x=xe1 yeze 3 includes physical coordinates such that x represents the coordinate of the bea axis y and z represents the coordinates along the cross section. he shape function atrix for the fullyparaetrized three-diensional bea eleent consisting of 4 degrees of freedo can be written as follows:

5 S( )= S1I SI S3I S4I S5I S6I S7I S8I (0) where the shape functions S1 S8 in the local eleent coordinates can be written as: 8 x ly lz x ly l S1 =1/4 1 S =1/8l 1 1 S3 = 1/4 1 S4 = 1/4 1 S5 = 1/4 1 S6 =1/8l 1 1 S7 =1/4 1 S =1/4 1 where the relations x / lx 1 y /ly and z /lz define the local eleent coordinates where l x ly and l z are diensions of the eleent size in directions x y and z. he kineatics of the eleent in the reference configuration at tie t =0 can be described as r= S ( xe ) where e = e (0). he vector e contains both translational and rotational coordinates of the eleent and it can be written at node i of the three-diensional fully-paraeterized eleent as follows: z (1) where the following notations for gradients are used: e () i () () () () = i i i i x y z r r r r () () i r 1 () i () i () = i r r r = ; = xyz (3) () i r 3 he elastic forces for the absolute nodal coordinate bea eleent can be defined by using threediensional elasticity or the elastic line approach. In the case of three-diensional elasticity the strains and stresses are defined using general continuu echanics where the variation of the strain energy with respect to the nodal coordinates can be written as Wint = : d V = : E S E d V V S e V e (4) where S is the second Piola-Kirchhoff stress tensor and E is the Green strain tensor. he vector of elastic forces can be identified fro Equation as follows: = : E F d. e S V V e (5) he second Piola-Kirchhoff stress tensor stress tensor can be written as S= ( tr E) I E (6) where I is the identity tensor and are ae's aterial coefficients. he Green strain tensor can be written as 1 E= ( F FI ) (7) where F is the deforation gradient tensor which can be presented in ters of the initial and current configurations r and r as follows:

6 r r r F= = r x x 1 (8) he variation of the kinetic energy can be presented as W = rrd V = e S S d V e (9) kin V V fro which the ass atrix of the eleent can be identified as follows: M= S S dv (30) V As can be concluded fro Equation (14) the ass atrix is constant as it is not a function of the nodal coordinates. his will save tie on coputation especially when an explicit tie integration ethod is used. It is well known that fully-paraetrized eleents where strains and stresses are defined by the general continuu echanics approach denoted by ANCFel in the study. However the bea eleents based on the general continuu echanics suffer fro different locking probles due to low-order approxiation in thickness directions. he elastic forces can be deterined by using an elastic line approach; see for exaple [10]. he elastic line approach is an alternative to general continuu echanics and all deforations are evaluated along the elastic line. he siilar ipleentation with [] denoted by ANCFel is used as the bea eleent based on the elastic line in this study. In this approach the shear locking is avoided by using linear interpolation for shear deforation. 4 COMPAISON OF FOMUAIONS Geoetrically exact and absolute nodal coordinate bea forulations are based on the continuu echanics and especially the principle of virtual work. he work conjugate pairs for the virtual internal work are chosen a different way: in the geoetrically exact bea forulation the pair is F:P where F is deforation tensor and P is the first Piola-Kirchhoff stress tensor. espectively in the absolute nodal coordinate forulation the pair is S: E where S the second Piola-Kirchhoff stress tensor and E is the Green strain tensor. However these work conjugare pairs are equivalent as is well known in the continuu echanics. Slight disparity arises when introducing constitute odels. he ain difference between forulations lies how the placeent is represented. In the geoetrically exact forulations the placeent is represented by the placeent of the bea center line and the spatial frae which represents the bea cross-section Eqn (). Moreover the rotation of the bea cross-section is given with aid of the rotation atrix which is interpolated by the total rotation vector () s. herefore the geoetrically exact bea forulation has three translational and three rotational degrees of freedo per node. Due to nonlinear character of the rotation the ass atrix has the state variable depencency via the rotation vector. Moreover this character arises the centrifugal stiffness and gyroscopic daping atrices that are nonsyetrical. However there nonsyetric velocity dependend atrices can be neglected in contentional engineering probles and in the iplicit tie integration schee syetric forulation can be utilized. he apply of the total rotion vector has notable benefits like inial nuber of variables no existence of variables with very high stiffness locking phenoenos can be eliinated rather easily. he use of the total rotation vector and the copleent rotation vector avoid the singularity proble of finite rotations under three-diensional rotations. In the case of ANCF eleents the singularity is avoided with a large nuber of general coordinates. 5 NUMEICA EXAMPES he nuerical exaples presented in this section deal with static and dynaic behavior of the previously studied bea eleents. In order to copare the theories nuerical exaples are solved with the bea eleents based on the geoetrically exact theory and the absolute nodal coordinate forulation which are denoted by GEBF [6] and ANCF; the latter eleent being introduced in []. As entioned in previous chapters the difference between these eleents is their description for rotation. In the GEBF the rotation is described with the total rotation vector and in the ANCF the coponents of

7 deforation gradients are used in description of rotation as well as deforation of the cross section. he both theories are presented in the total agrangian forulation so approxiations for angles or corotational fraes are not eployed in the forulations. he extra attention is given to the convergence of the eleents under torsion and coputational effort in the following nuerical section. he elastic forces of the ANCF eleents are integrated using 4-point quadrature in the -direction and -point quadrature in the and directions. In the GEBF eleents the linear interpolation is used for the rotation and translation fields and 1-point quadrature for the elastic forces. 5.1 arge deforation cantilever bea A cantilever bea with a rectangular cross section (Figure 1) is studied in the following section to show the convergence properties of the studied eleents under the influence of large deforation. he paraeters of the cantilever bea are the length = 10 the applied force F = N and the bending oent M =.5 N. A quiet siilar cantilever proble was studied in [4]. In the cantilever bea the torsional rigidity GJ = 00 is given in order to find paraeters for voluetrically interpolated ANCF eleents. he proble is solved with GEBF ANCFel and ANCFc eleents. In the able 1 the converged solutions within four digits are shown. he nuber of eleents needed for the converged solution in four digits is shown in the parenthesis. he reference converged solution is obtained by using linear ANSYS bea 188 eleent (which is based on a geoetrically exact bea theory). Figure 1. Cantilever bea with an applied force and oent and its proble data. Eleent N (DOFs) u x -displaceent u y -displaceent u z -displaceent GEBF 64 (384) ANCFc 3 (386) ANCFel 16 (194) ANSYS Bea (384) ANSYS Bea 188 (quadratic) 8 (96) able 1. ip displaceent of the cantilever bea under the applied force and the oent (the nuber of eleents is given in parenthesis). As seen in able 1 the converged solution within four digits of GEBF and ANCFel coincide quite well with the reference solution. In the exaple the Poisson value is given as zero and therefore the ANCFc eleent does not show Poisson locking phenoena. As can be seen fro converged solution in able 1 and convergence rate for the transverse displaceent of ANCFc eleent in Figure the ANCFc eleent suffer fro nuerical locking leading to the clearly different displaceents. his shear locking is avoided in ANCFel eleent where shear deforations are linearized. Only 16 ANCFel eleents (194 DOFs) are needed for the converged results whereas 64 GEBF eleents (384 DOFs) are needed. However the ANCFel eleent still leads to non-onotonic convergence rate. he convergence rate of the GEBF eleent follows the quadratic convergence rate. It shall be noted that only one GEBF eleent is needed for the converged solution in the case of large rotation pure torsion exaple in [13] while the studied ANCF eleents need 4-5 eleents for the converged solution.

8 Figure. Convergence rate of the studied eleents. he relative error in y-displaceent at the free end of the cantilever bea is considered. he dashed lines illustrate the quadratic convergence rate. 5. he dynaics of a flexible free bea undergoing large otion In this section dynaics of the studied bea eleents are studied under the influence of large displaceents. he exaple introduced in [1] is known as the Flying Spaghetti proble and was used for verification of eissner's shear deforable bea eleent based on the large rotation vector approach [3]. he proble data is shown below in Figure 3. Figure 3. Flexible free bea with an applied force and oent and its proble data. A siilar nuerical test is used for the verification of the absolute nodal coordinate forulation [3 5]. As entioned in [3] the Flying Spaghetti proble is favored for verification of dynaics of the beas because all of the paraeters in elastic forces and inertia are sensitive. he Flying Spaghetti proble is solved with the GEBF and the ANCFel eleents. he reference solution is deterined with ANSYS Bea 188 presented in [3]. he tie integration for the equations of otion is obtained with the trapezoidal integrator ode3t in MAAB. he values and 0.01 for the absolute and relative tolerances are used. If the ordinary differential syste equations (ODE) have a ixture of fast and slow changing variables as it is in the case the ODE syste is nuerically stiff. For such ODEs stiff deterinistic solvers are the best choice. he ode3t solver is an ipleentation of the trapezoidal rule. However the trapezoidal rule can effectively be utilized if the proble is only oderately stiff and a solution is needed without nuerical daping. It is known that a choice of paraeters of Newark ethod gives the trapezoidal rule. he Newark ethod is the best-known tie integration ethod in the structural dynaics.

9 Eleent N (DOFs) tie steps evaluations GEBF 3 (198) GEBF 64 (390) ANCFel 16 (04) ANCFel 3 (396) ANCFc 16 (04) ANCFc 3(396) able. Coputational effort. Nubers of eleents degrees of freedo (DOFs) tie steps and function evaluations of the differential equation are given Figure 4. Vertical displaceent of point B with different bea eleents. As can be seen fro Figure 4 the solution of the ANCFel is not converged. he converged solution is not reached due to the large coputation effort (able ) whereas solution by 3 GEBF eleents corresponds the reference solution given by ANSYS Bea 188. he solution of ANCFc is not shown due to the strong effect of the shear locking. Based on the coparison the ANCF is not coputationally effective in this special exaple of large deforation dynaics. 6 CONCUSION In the absolute nodal coordinate forulation the rotation and deforation of the cross section and centerline are accounted for using the position gradient vectors. he use of gradient vectors needs the Heritean type of interpolation leading to the constant ass atrix that can be coputationally efficient in the case of explicit integrators. In coputational point of view another benefit in the absolute nodal coordinate forulation is to avoid the singularity in three diensional rotations using larger set of generalized coordinates than in the bea theories. In the total agrangian forulation the singularity proble can be avoided by varying paraeterization on the rotation anifold. his approach is used in the geoetrically exact forulation in this study. In the study the coputation efforts of different forulations are considered throughout large deforation static and dynaic exaples. Based on the solved nuerical exaples the ipleented ANCF bea eleents are not as coputationally efficient as the ipleented singularity-free GEBF eleent. he inefficiency of the ANCF eleents can be occurred fro different nuerical locking and fro inappropriate description for the torsion and bending. herefore in the future work the inefficiency of ANCF bea eleents has to be solved in order to reach the efficiency of GEBF eleent. Additionally

10 ore different benchark probles are needed for the objective efficiency coparison between these nonlinear finite eleent forulations. Further study is still required. In the dynaic exaples tie integration ethods for stiff systes should be tested. As discussed before the exploited trapezoid tie integration rule is efficient for only oderately stiff probles. Moreover different fully three-diensional dynaic probles should be carefully studied. In addition the geoetrically exact bea eleent with higher order interpolation should be investigated. It ight expect a better convergence rate than for linearly interpolated eleents. EFEENCES [1] CADONA A. AND GÉADIN M. A Bea Finite Eleent Non-inear heory with Finite otations. International Journal for Nuerical Methods in Engineering 6 (1988) [] DUFVA E. K. SOPANEN J.. AND MIKKOA A. M. A hree-diensional Bea Eleent Based on a Cross-Sectional Coordinate Syste Approach. Nonlinear Dynaics 43 4 (005) [3] GESMAY J. MAIKAINEN M. K. AND MIKKOA A. M. A Geoetrically Exact Bea Eleent Based on the Absolute Nodal Coordinate Forulation. Multibody Syste Dynaics 0 4 (008) [4] IBAHIMBEGOVI A. FEY F. AND KOZA I. Coputational Aspects of Vector-ike Paraetrization of hree-diensional Finite otations. International Journal for Nuerical Methods in Engineering 38 (1995) [5] MAIKAINEN M. K. VON HEZEN. MIKKOA A. M. AND GESMAY J. Eliination of High Frequencies in the Absolute Nodal Coordinate Forulation Journal of Multi-body Dynaics Proceedings Part K 4 1 (010) [6] MÄKINEN J. otal agrangian eissner s Geoetrically Exact Bea Eleent without Singularities. International Journal for Nuerical Methods in Engineering 70 9 (007) [7] MÄKINEN J. A Forulation for Flexible Multibody Mechanics agrangian Geoetrically Exact Bea Eleents Using Constraint Manifold Paraetrization. U Applied Mechanics and Optiization esearch eport 004:3 89 pp. U: [8] EISSNE E. On One-Diensional arge-displaceent Finite-Strain Bea heory Studies in Applied Matheatics 5 (1973) [9] OMEO I. A Coparison of Finite Eleents for Nonlinear Beas: he Absolute Nodal Coordinate Forulation and Geoetrically Exact Forulations. Multibody Syste Dynaics 0 1 (008) [10] SCHWAB A.. AND MEIJAAD J. P. Coparison of hree-diensional Flexible Bea Eleents for Dynaic Analysis: Finite Eleent Method and Absolute Nodal Coordinate Forulation. In Proceedings of the IDEC/CIE 005 ASME 005 International Design Engineering echnical Conferences & Coputers and Inforation in Engineering Conference Paper Nuber DEC (ong Beach (CA) USA 4-8 Septeber 005). [11] SHABANA A. A. Definition of the Slopes and the Finite Eleent Absolute Nodal Coordinate Forulation. Multibody Syste Dynaics 1 (1997) [1] SIMO J.C. AND VU-QUOC. On the Dynaics in Space of ods Undergoing arge Motion - A Geoetrically Exact Approach Coputer Methods in Applied Mechanics and Engineering 66 (1988) [13] SIMO J.C. FOX D. D. AND IFAI M. S. On a Stress esultant Geoetrically Exact Shell Model. Part III: Coputational Aspects of the Nonlinear heory. Coputer Methods in Applied Mechanics and Engineering 79 (1990) 1-70.