on complex lamellar motion of a prim gas an intrinsic formulation of prim s canonical equations of gas dynamics is employed to establish a generalization of crocco s theorem to complex lamellar motion this new result is then employed to establish the existence of a heisenberg spin type equation in such motion the optimal coordinate systems in the generalized lagrangian method which are known to exist only in complex lamellar motion are then shown by intrinsic geometric methods to correspond to a marching direction along geodesics on generalized bernoulli surfaces

modelling of superelements in mechanism analysis the paper deals with substructuring for dynamic analysis of flexible multibody systems three different techniques based on component synthesis are discussed corresponding respectively to fully consistent mass discretization lumped mass discretization and corotational approximation of inertia forces to simplify the computer implementation only the lumped mass and corotational approximations have been considered in detail and programmed both approaches are validated on simple examples of rotating beams for which a full elastic model is available using a fully non linear beam element the computational efficiency of the corotational inertia approach is also demonstrated on the deployment of a large flexible satellite antenna

an element free galerkin approach for rigidflexible coupling dynamics in 2d state based on an improved element free galerkin efg method and the geometric nonlinear elastic theory an element free numerical approach for rigidflexible coupling dynamics of a rotating hubbeam system is implemented using the hamilton principle the coupled nonlinear integral differential governing equations are derived in which the dynamic stiffening effect is expressed by the longitudinal shrinking of the beam induced by the transverse displacement by the efg method where the global interpolating moving least squares imls and generalized moving least squares igmls were used for discretizing the longitudinal and transverse deformation variables respectively the spatially element free discretized dynamic equations of the system are obtained and the newmark scheme was selected as the time integration method for the numerical computation the superiority of the global interpolating property makes the ease for imposing both displacement and derivative boundary conditions the transverse bending vibration and structural dynamics of the flexible beam in non inertial system are analyzed in the numerical simulations and influence parameters of the efg method are also fully discussed simulation results verify the feasibility of the improved efg approach for rigidflexible coupling dynamics

non linear substructure approach for dynamic analysis of rigid flexible multibody systems abstract this paper presents a substructure synthesis method ssm for non linear analysis of multibody systems the detailed derivation of the equation of motion which takes into account the geometric non linear effects of large rotation undergoing small strain elastic deformation is presented using the substructure synthesis approach the equation of motion of the systems can be simplified for rigid and flexible substructures for the flexible substructure the equation of motion is condensed through the boundary conditions at the interface between the flexible and rigid substructures as a result equations of motion for multi flexible body systems including the geometric non linear effects of large rotation are derived to demonstrate the applicability and accuracy of the proposed approach an example of a two link manipulator was chosen for this presentation the results using the linear and non linear models are presented to highlight the effects of geometric non linearities

explicit free floating beam element summary a two node free floating beam element capable of undergoing arbitrary large displacements and finite rotations is presented in explicit form the configuration of the beam in three dimensional space is represented by the global components of the position of the beam nodes and an associated set of convected base vectors directors the local constitutive stiffness is derived from the complementary energy of a set of six independent deformation modes each corresponding to an equilibrium state of constant internal force or moment the deformation modes are characterized by generalized strains formed via scalar products of the element related vectors this leads to a homogeneous quadratic strain definition in terms of the generalized displacements whereby the elastic energy becomes at most bi quadratic additionally the use of independent equilibrium modes to set up the element stiffness avoids interpolation of kinematic variables resulting in a locking free formulation in terms of three explicit matrices a set of classic benchmark examples illustrates excellent performance of the explicit beam element copyright 2014 john wiley sons ltd

on a new defect of shock capturing methods this paper proposes an explanation and a cure or avoidance to the new defect found of eulerian shock capturing methods in a note on the conservative schemes for the euler equations by tang and liu h tang tiegang liu a note on the conservative schemes for the euler equations j comput phys 218 2006 451 459 the latter gives a numerical investigation using several popular high resolution conservative schemes applied to riemann problems of inviscid compressible perfect gas flows in eulerian and lagrangian coordinates with an initial high density ratio as well as a high pressure ratio the results show that these methods work very inefficiently when applied to such problems and may give inaccurate numerical results especially in shock location or speed even with a very fine grid we have found that in problems of this type a strong rarefaction wave srw is present adjacent to a contact line godunov averaging over the wave then produces large errors which when the wave is strong also persist for a long time the cumulative error is thus very large which violates the strength of the contact line adjacent to it which in turn affects the speed and hence the location of the shock on the other side of the contact we confirm this numerically using a method based on the unified coordinates with the shock adaptive godunov scheme plus contact strength preserving the method when applied to the examples 2 1 and 2 2 of tang and liu h tang tiegang liu a note on the conservative schemes for the euler equations j comput phys 218 2006 451 459 produces high quality results even for comparatively coarse grids

a new lagrangian method for time dependent inviscid flow computation this paper presents a new lagrangian approach for the two dimensional 2 d time dependent euler equations it may be considered as a sequel to the authors previous lagrangian approaches for steady supersonic flow computations c y loh and w h hui j comput phys 89 1990 pp 207 240 w h hui and c y loh j comput phys 103 1992 pp 450 464 w h hui and c y loh j comput phys 103 1992 pp 465 471 c y loh and m s liou j comput phys 104 1993 pp 150 161 c y loh and m s liou siam j sci comput 15 1994 pp 1038 1058 c y loh and m s liou j comput phys 113 1994 pp 224 248 the theoretical background and the intrinsic flow coordinates as well as the lagrangian conservation form are introduced based on the concept of material functions or path functions a tvd scheme of the godunov type is chosen to describe the numerical procedure several examples are then given to justify the claimed advantages of the new methodology namely a any contact discontinuities are crisply solved and b grids are automatically and accurately generated following pathlines

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

non linear theory for elastic beams and rods and its finite element representation abstract in terms of displacement components and angular coordinates expressions are given for large curvatures of elastic beams and rods the application of these expressions to bifurcation and stability problems is discussed on the basis of the principle of virtual work this approach is compared with the finite element representation given in a previous paper 1 an error in this previous paper is corrected further it is shown that the condition of zero extension of the axis in the case of the euler column requires a division into a rather large number of finite elements to ensure an accurate result for the stability coefficient the conditions of zero extension of the axis and zero curvature about one principal axis of the cross section in the case of the lateral buckling of the end loaded cantilever have similar consequences

analysis of flexible multibody systems with spatial beams using mixed variational principles a general nite element formulation is presented for dynamic analysis of spatial elastic beams for small strains in a multi body conguration the tangent maps associated to the nite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form a corotational method and a mixed variational method are used to compute the tangent stiness matrix the tangent constraint matrices are obtained using consistent linearization of the constraint equations the tangent inertia matrices including the gyroscopic and centrifugal terms are also obtained by using the tangent maps of rotation the numerical examples analyzed in this paper include dynamic analysis of exible beam structures and multi exible body systems with open and closed kinematic loops a comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used 1998 john wiley sons ltd this paper deals with the modelling of the spatial dynamic behaviour of homogeneous isotropic and linear elastic one dimensional deformable bodies such as beams or rods undergoing arbitrarily large rotations and translations and small strains the beam may be connected to other beams by means of kinematic constraints to form an open or closed loop multi exible body mechanical system a previous paper 1 of which the present one is a sequel has investigated two dierent methods to compute the tangent stiness of the elastic beam in particular the elastic tangent stiness matrix and residual vector for a space beam were derived with two dierent approaches a primal total lagrangean tl corotational approach and a mixed updated lagrangean ul approach the idea of that derivation was to formally adopt a variational setting to derive the eld equations of motion of constrained exible bodies previously a consistent variational approach for multi exible body dynamics with beams has only been presented by cardona and geradin 2 in their work they used the principle of virtual work and methods of non linear structural

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

a shock adaptive godunov scheme based on the generalised lagrangian formulation application of the godunov scheme to the euler equations of gas dynamics based on the eulerian formulation of flow smears discontinuities sliplines especially over several computational cells while the accuracy in the smooth flow region is of the order 0 h where h is the cell width based on the generalised lagrangian formulation glf of hui et al the godunov scheme yields superior accuracy by the use of coordinate streamlines in the glf the slip line itself a streamline is resolved crisply infinite shock resolution is achieved through the splitting of shock cells an improved entropy conservation formulation of the governing equations is also proposed for computations in smooth flow regions finally the use of the glf substantially simplifies the programming logic resulting in a very robust accurate and efficient scheme 15 refs 6 figs

on convergence in elliptic shape optimization the present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization motivated by illustrative examples an abstract setting of the underlying shape optimization problem is suggested taking into account the so called two norm discrepancy a ritz galerkin type method is applied to solve the associated necessary condition existence and convergence of approximate solutions are proved provided that the infinite dimensional shape problem admits a stable second order optimizer the rate of convergence is confirmed by numerical results

a new formulation for the direct dynamic simulation of flexible mechanisms based on the newton euler inverse method this paper is concerned with the direct dynamic analysis of mechanisms comprising flexible members it is first recalled how the time integration of the dynamic equations governing the motion of a flexible mechanism is conditioned by the availability of the so called iteration matrix the latter is obtained not only from the mass matrix which is sufficient in the rigid case but also from the stiffness and damping tangent matrices hence in a context based upon relative co ordinates where the deformation of the flexible bodies is described by component modes two formulations a classical one and a new one which permit the determination of this iteration matrix are presented the classical formulation is based upon the lagrange s theorem the new one consists in a direct calculation of the residues of the dynamic equations by using the inverse method of newton euler in this case the iteration matrix is obtained by a numerical derivation technique which proves well conditioned a first simple example is treated in order to compare the two formulations a more complex one illustrates the real capabilities of the new formulation

co rotational finite element formulation for thin walled beams with generic open section a consistent co rotational total lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin walled beams with generic open section is presented the element developed here has two nodes with seven degrees of freedom per node the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis the deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element the element nodal forces are derived using the virtual work principle the virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces a procedure is proposed to determine the virtual rigid body motion the way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent in element nodal forces all coupling among bending twisting and stretching deformations of the beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory in the derivation of the element tangent stiffness matrix the change of element nodal forces induced by the element rigid body rotations should be considered for the present method thus a stability matrix is included in the element tangent stiffness matrix an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state numerical examples are presented to investigate the accuracy and efficiency of the proposed method the effect of the terms in the element nodal force and tangent stiffness matrix which will converge to zero with the decrease of element size on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples 2005 elsevier b v all rights reserved

a spatially curved beam element with warping and wagner effects this paper presents a new spatially curved beam element with warping and wagner effects that can be used for the non linear large displacement analysis of members that are curved in space the non linear behaviour of members curved in space shows that the wagner effects are substantial in the large twist rotation analysis most existing finite beam element models such as abaqus and ansys cannot predict the non linear large displacement response of members curved in space correctly because the wagner effects viz the wagner moment and the corresponding finite strain terms have not been considered in these finite beam elements as a consequence these finite beam elements do not provide correct predictions for the out of plane buckling and postbuckling behaviour of arches as well in this paper the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements the warping and wagner effects both the wagner moment and the corresponding finite strain terms and their constitutive relationship are included in the spatially curved beam element two components of the initial curvature the initial twist and their interactions with the displacements are also considered in the spatially curved beam element this ensures that the large twist rotation analysis for the members curved in space is accurate comparisons with existing experimental analytical and numerical results show that the spatially curved beam element is accurate and efficient for the non linear elastic analysis of curved members buckling and postbuckling analysis of arches and in its ability to predict large deflections and twist rotations in more arbitrarily curved members copyright 2005 john wiley sons ltd

a unified coordinate system for solving the three dimensional euler equations it is well known that the use of eulerian coordinates for shock capturing methods results in badly smeared slip lines and that lagrangian coordinates while capable of producing sharp slip line resolution may result in severe grid deformation causing inaccuracy and even breakdown of computation a unified coordinate system is introduced in which the flow variables are considered to be functions of time and of some permanent identification of pseudo particles which move with velocity hq q being the velocity of fluid particles it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 for two dimensional inviscid flow the free function h is chosen so as to preserve the grid angles this results in a coordinate system which avoids excessive numerical diffusion across slip lines in the eulerian coordinates and avoids severe grid deformation in the lagrangian coordinates yet it retains sharp resolution of slip lines especially for steady flow furthermore the two dimensional unsteady euler equations of gasdynamics in the unified coordinates are found to be hyperbolic for all values of h except when h 1 i e lagrangian in the latter case the euler equations are only weakly hyperbolic lacking one eigenvector although all eigenvalues are real the consequences of this deficiency of the lagrangian coordinates are pointed out in connection with numerical computation

four node semi eas element in six field nonlinear theory of shells we propose a new four node c finite element for shell structures undergoing unlimited translations and rotations the considerations concern the general six field theory of shells with asymmetric strain measures in geometrically nonlinear static problems the shell kinematics is of the two dimensional cosserat continuum type and is described by two independent fields the vector field for translations and the proper orthogonal tensor field for rotations all three rotational parameters are treated here as independent hence as a consequence of the shell theory the proposed element has naturally six engineering degrees of freedom at each node with the so called drilling rotation this property makes the element suitable for analysis of shell structures containing folds branches or intersections to avoid locking phenomena we use the enhanced assumed strain eas concept we derive and linearize the modified hu washizu principle for six field theory of shells what makes the present approach original is the combination of eas method with asymmetric membrane strain measures based on literature we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry to gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches

generalized mixed finite element model for pre and post quasistatic buckling response of thin walled framed structures adopting an updated lagrangian approach the general framework for the fully non linear analysis of thin walled framed structures is developed using a simple two node c0 model hmb2 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of modified hellinger reissner type with independent assumptions for displacement and strain fields all coupled significant modes of deformations i e stretching bending shear torsion and warping are accounted for in the generalized beam theory employed emphasis is placed on devising effective solution procedures to deal with finite rotations in space particularly with regard to their effect on the derivations of load correction matrices corresponding to configuration dependent externally applied forces moments the effectiveness and practical usefulness of present model are demonstrated through a number of test problems involving beam assemblages undergoing large displacements and rotations in space

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

dynamics of flexible multibody systems using cartesian co ordinates and large displacement theory a formulation for the dynamic analysis of flexible systems composed of slender bodies that can be accurately modelled by beams is presented in this paper a new set of state variables composed of cartesian co ordinates of points and unit vectors is introduced to define the beam with respect to an inertial frame a non linear timoshenko beam finite element capable of handling finite displacements with small linear elastic strains is developed this allows relative displacements between material points of a single beam to be arbitrarily large since deformations are not explicit variables there is no need to define a moving reference frame attached to each flexible body instead deformations are obtained through a displacement deformation relation based on finite displacement beam theory the differential equations of motion are obtained using the lagrange equations a symmetric constant and sparse mass matrix is obtained in the inertial frame constraints are introduced with a penalty formulation and the resulting set of ordinary differential equations is integrated with newmark s family of methods the whole formulation is extremely simple and the results demonstrate the capabilities and efficiency of the proposed method for dynamic simulation even when relative displacements are finite in a single beam or coupling effects are significant

dynamics of rectangular plates undergoing prescribed overall motion a set of linear equations of motion for rectangular plates undergoing prescribed overall motion is derived in this paper two in plane stretch variables with which the in plane strain energy of the plate can be expressed in an exact quadratic form are employed and approximated to derive the equations of motion the equations of motion include motion induced stiffness variation terms which are expressed as explicit functions of the prescribed overall motion the effects of the motion induced stiffness variation terms which are neglected in the conventional linear modelling method on the dynamic response of the plate are investigated the reliability and the accuracy of the equations of motion which are derived in the present work are examined through the numerical study

a crystalline motion uniqueness and geometric properties the author studies a model of crystalline motion in the plane existence and uniqueness of local in time solutions are shown geometric properties of the flow are studied assuming that the wulff shape is a regular n sided polygon the author shows that a small convex grain of ice immersed in the cold melt shrinks provided that the flow does not overly deform the initial interface the author is able to verify the last condition only if the initial interface is a scaled wulff shape in this case the free boundary will remain a scaled wulff shape at later times this is shown by using an isoperimetric inequality

a simple element for static and dynamic response of beams with material and geometric nonlinearities a unified simple 6 degrees of freedom beam finite element and the associated computational procedures have been developed for the fast and efficient solution of a wide class of static and dynamic response problems of the beam type with material and or geometrical nonlinearities the material nonlinearity is treated by including its effect in the governing equations by forming the stiffness matrix of each element using a two dimensional grid of gauss points and using the material properties at each point corresponding to the uniaxial strain at that point examples are provided for metal and reinforced concrete beams a powerful yet straightforward method for the solution of elastica problems of beams and frames using the beam element developed by the senior author has been extended for determining the dynamic response of beams undergoing large displacements including large rotations the solution procedure involves piecewise linearization of response equations and iterations at each incremental step to achieve equilibrium the solution procedure is simple and easy to apply a variety of problems is solved to determine the applicability of the proposed simple formulations excellent agreement with existing analytical solutions which employ higher order elements demonstrates the efficiency and versatility of the present simple beam element in nonlinear analysis

direct computation of critical equilibrium states for spatial beams and frames summary in this paper explicit formulas for second order derivatives of the residual vector with respect to the state variables for a geometrically exact 3d beam element based on the reissner s model are presented these derivatives are required when a direct non linear stability eigenvalue problem is solved by the newton s method if the external load is parametrized by a single parameter such an eigenvalue problem consists of solving the critical state variables the eigenmode and the critical load parameter from the equation system consisting of the equilibrium equations the criticality condition and some auxiliary conditions depending on the type of a critical point copyright 2011 john wiley sons ltd

a general formulation in rigid multibody dynamics the dynamics of rigid multibodies is traditionally formulated by means of either minimal or redundant co ordinates methods an alternative approach is here proposed whereby a highly redundant set of co ordinates is adopted as a result the equations of motion of the constrained bodies are decoupled several meaningful parameters are directly available and the constraint conditions are enforced in a very natural way the first part of the paper presents the basic meanings and the theoretical developments of the formulation the second develops a numerical approximation for the methodology proposed in the first part the non linear system of differential algebraic equations governing the motion of the multibody is reduced to its weak form it is linearized by applying a newton raphson procedure and approximated through the method of finite elements in time the details of the numerical application of this method are discussed and a solution procedure is presented finally some numerical examples involving tree and closed loop topologies prove the capability of the present formulation in handling multibody dynamics

long term stable time integration scheme for dynamic analysis of planar geometrically exact timoshenko beams abstract in this paper an energy momentum method for geometrically exact timoshenko type beam is proposed the classical time integration schemes in dynamics are known to exhibit instability in the non linear regime the so called timoshenko type beam with the use of rotational degree of freedom leads to simpler strain relations and simpler expressions of the inertial terms as compared to the well known bernoulli type model the treatment of the bernoulli model has been recently addressed by the authors in this present work we extend our approach of using the strain rates to define the strain fields to in plane geometrically exact timoshenko type beams the large rotational degrees of freedom are exactly computed the well known enhanced strain method is used to avoid locking phenomena conservation of energy momentum and angular momentum is proved formally and numerically the excellent performance of the formulation will be demonstrated through a range of examples

co rotational formulation for geometric nonlinear analysis of doubly symmetric thin walled beams abstract a doubly symmetric thin walled beam element with open section is derived using co rotational cr total lagrangian tl formulation the effects of deformation dependent third order terms of element nodal forces on the buckling load and post buckling behavior are investigated all coupling among bending twisting and stretching deformations for beam element is considered by consistent second order linearization of the fully geometrically nonlinear beam theory however all third order terms of nodal forces which are relevant to the twist rate rate of twist rate and curvature of the beam axis are also considered an incremental iterative method based on the newton raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations the zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state a parabolic interpolation method of the arc length is used to find the buckling load numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third order terms of element nodal forces on the buckling load and post buckling behavior of doubly symmetric thin walled beams

computation of the shallow water equations using the unified coordinates two general coordinate systems have been used extensively in computational fluid dynamics the eulerian and the lagrangian the eulerian coordinates cause excessive numerical diffusion across flow discontinuities slip lines in particular the lagrangian coordinates on the other hand can resolve slip lines sharply but cause severe grid deformation resulting in large errors and even breakdown of the computation recently in the spirit of the arbitrary lagrangian eulerian ale approach w h hui p y li and z w li j comput phys 153 1999 pp 596 637 have introduced a unified coordinate system which moves with velocity hq q being the velocity of the fluid particle it includes the eulerian system as a special case when h 0 and the lagrangian when h 1 and was shown for the two dimensional euler equations of gas dynamics to be superior to both eulerian and lagrangian systems the main purpose of this paper is to adopt this unified coordinate system to solve the shallow water equations it will be shown that computational results using the unified system are superior to existing results based on either the eulerian system or lagrangian system in that it a resolves slip lines sharply especially for steady flow b avoids grid deformation and computation breakdown in lagrangian coordinates and c avoids spurious flow produced by lagrangian coordinates

nonlinear aeroelastic study for folding wing structures doi 10 2514 1 44868 a folding wing structure consisting of three components a fuselage an inboard wing and an outboard wing is modeled computationally using a geometrically nonlinear structural dynamics theory based upon von karman strainsandathree dimensionalvortexlatticeaerodynamicmodelwithanexacttangent flowboundaryconditionand planar wake assumption the structural dynamic equations of motion are discretized in space using a discrete ritz basis derived from finite element analysis and component synthesis results from the computational model are comparedwiththosefromexperimentsdesignedandtestedinthedukeuniversitywindtunnelforthreefoldingwing configurations stablelimitcycleoscillationsat flowvelocitiesbeyondthelinear fluttervelocityaremeasuredinwindtunnel experiments and predicted using the computational model overall the limit cycle oscillation magnitude and dominant response frequency results from theory show good agreement with those measured in the experiment qualitatively both the experimental and theoretical limit cycle oscillation curves for the inboard wing show limited nonlinear stiffening with flow velocity for the range of velocities tested the theoretical model also predicts that the outboard wing limit cycle oscillation tip displacements for the folding wing configuration with the largest outboard folding angle is significantly higher than the two other configurations unlike the inboard wing for each configuration the outboard wing theoretical limit cycle oscillation curves do show a trend that is reminiscent of a stiffening nonlinearity

a procedure for analysing space frames with partial warping restraint stiffness matrices for three dimensional beam elements that include the effect of warping restraint on elastic torsional response have been derived by various investigators using one of the available stiffness matrices and assuming that the warping boundary conditions can be specified on a member by member basis an elastic warping support is introduced to represent conditions of partial warping restraint at the member ends the concept of a warping indicator is then introduced to facilitate use of warping springs following this static condensation is used to eliminate the restrained warping degrees of freedom the condensed stiffness matrices for the elements can then be assembled to yield a global stiffness matrix in the global matrix continuous warping degrees of freedom that is those internal to a member represented by several elements are expressed in local co ordinates the remainder are expressed in global co ordinates in the force recovery phase it is shown that an indirect method yields most accurate results for the bimoment and warping torsion when the twist function is represented by a cubic polynomial solutions to examples of linear elastic analysis are compared with well known analytical solutions to demonstrate the application of the method

hyperbolicity and optimal coordinates for the three dimensional supersonic euler equations the two dimensional 2 d steady euler equations for supersonic flow in cartesian coordinates x y is hyperbolic except on the singular line u a where a is the speed of sound and u is the x component of velocity however the cauchy problem for x x circ with initial data prescribed at x x circ is well posed only when u a for the 2 d case the optimal coordinates have been found w h hui and d l chu comput fluid dynamics 4 1996 pp 403 426 to be the orthogonal system consisting of streamlines and their orthogonal lines this gives the most robust and accurate computation for supersonic flow it is commonly thought e g c y loh and m s liou j comput phys 113 1994 pp 224 248 that the system of three dimensional 3 d steady euler equations for supersonic flow is hyperbolic the present paper shows that it is hyperbolic only when the velocity in the marching direction is supersonic in which case it also simultaneously guarantees the well posedness of the cau

a computational framework for the regularization of adjoint analysis in multiscale pde systems this paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale pde systems regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis typically l2 brackets are used in the definition of the cost functional the adjoint operator and the cost functional gradient if instead we adopt the more general sobolev brackets the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically the present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework the regularization strategies proposed are exemplified using a id kuramoto sivashinsky forecasting problem and computational examples are provided which exhibit their utility a multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed

primal and mixed variational principles for dynamics of spatial beams a general formulation is presented for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains the tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate implicitly the equations of motion in descriptor form a total lagrangian primal corotational method and an updated lagrangian mixed variational method are proposed to compute the tangent stiffness matrix the tangent inertia matrices including the gyroscopic and centrifugal terms are also obtained by using the tangent maps of rotation the numerical examples analyzed in this paper include static pre and postbuckling and dynamic analysis of flexible beams structures the new finite elements show a very good performance in terms of fewer number of elements used and accuracy during the simulation both for static and dynamic problems

an inverse model for a free boundary problem with a contact line steady case this paper reformulates the two phase solidification problem i e the stefan problem as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to pde constraints an advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase it is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated we describe an efficient iterative solution method for the stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional performance of the proposed approach is illustrated with sample computations concerning 2d steady solidification phenomena

geometric non linear substructuring for dynamics of flexible mechanical systems a substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non linear effects large relative displacement is permitted between points within bodies that undergo small strain elastic deformation components are divided into substructures on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure compatibility conditions are derived and imposed as constraint equations at boundary points between substructures system equations of motion that include geometric non linear effects of large rotation in terms of generalized co ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure are assembled the method is implemented in an industry standard flexible multibody dynamics code with minimal modification use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

deformation modes in the finite element absolute nodal coordinate formulation abstract the objective of this study is to provide interpretation of the deformation modes in the finite element absolute nodal coordinate formulation using several strain definitions in this finite element formulation the nodal coordinates consist of absolute position coordinates and gradients that can be used to define a unique rotation and deformation fields within the element as well as at the nodal points the results obtained in this study clearly show cross section deformation modes eliminated when the number of the finite element nodal coordinates is systematically and consistently reduced using the procedure discussed in this paper one can obtain a reduced order dynamic model eliminate position vector gradients that introduce high frequencies to the solution of some problems achieve the continuity of the remaining gradients at the nodal points and obtain a formulation that automatically satisfies the principle of work and energy furthermore the resulting dynamic model unlike large rotation finite element formulations leads to a unique rotation field and as a consequence the obtained formulation does not suffer from the problem of coordinate redundancy that characterizes existing large deformation finite element formulations in order to accurately define strain components that can have easy physical interpretation a material coordinate system is introduced to define the material element rotation and deformation using the material coordinate system the timoshenko reissner and euler bernoulli beam models can be systematically obtained as special cases of the absolute nodal coordinate formulation beam models while a constraint approach is used in this study to eliminate the cross section deformation modes it is important to point out as mentioned in this paper that lower order finite elements some of which already presented in previous investigations can be efficiently used in thin and stiff structure applications

formulation and solution of inverse spaghetti problem application to beam deployment dynamics a finite element model of the dynamics of axially moving highly flexible beams is presented the model is based on a geometrically nonlinear beam formulation that allows for large overall motions to account for the varying length of a deploying beam a moving finite element reference grid is incorporated within the nonlinear beam formulation such that the number of finite element nodes remains fixed and the finite element length is allowed to vary hamilton s law is used to formulate the equations of motion and a transient integration solution procedure is derived from a space time finite element discretization of the hamiltonian variational principle computational results of the methodology are presented for a planar inverse spaghetti problem he dynamics of axially moving beamlike structures are becoming increasingly important for the analysis of spacecraft and large space structures that deploy flexible appendages such as antenna stabilizing booms solar arrays and long trusslike structures as well as for other applications such as magnetic tape drives printing machines traveling cables and band saws an extensive amount of research has focused on the modeling of axially moving continua to analyze the dynamic behavior of such structures one of the most common models of axially moving continua that has received the most attention is the traveling string problem in what he termed the spaghetti problem carrier1 analyzed the motion of a string being accelerated upward into a fixed orifice in other works since then analytical expressions representing both linear and nonlinear vibrations of axially moving strings subject to various support conditions have been derived 2 9 the dynamics of beamlike structures traveling between or over fixed supports have also been studied to address industrial applications such as high speed tape drives and band saws 10 13 to address spacecraft applications in which antenna or other long flexible structures are deployed from a host satellite the dynamics of a cantilever beam that is ejected from a single fixed guide are modeled using bernoulli euler beam theory in ref 14 in a quite different approach the extrusion of a beam from a rotating base is modeled using a series of elastically connected rigid links in which the number of rigid links are continuously increased decreased to account for deployment retrieval of the beam 15 general formulations of axially moving bernoulli euler beam models cantilevered to a rotating rigid host body have also been developed to study the effect of appendage deployment on the attitude dynamics of orbiting spacecraft 16 20 all of these given analyses are based on an assumption of small elastic deformations with the exception of the discrete spring mass model of ref 15 all of the other models that have been referenced thus far use linear combinations of space depende nt admissible functions of a time varying length weighted by time dependen t generalized coordi

finite element formulation of geometrically exact three dimensional beam theories based on interpolation of strain measures this paper presents a new finite element formulation of the geometrically exact finite strain beam theory the governing equations of the beam element are derived in which the strain vectors are the only unknown functions the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal is enforced to be satisfied at chosen points the solution is found by a collocation algorithm the linearity of the strain space not only simplifies the application of newton s method on the non linear configuration space but also leads to the strain objectivity of the proposed method the accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples c 2003 elsevier b v all rights reserved

vortex design problem in this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity the problem is formulated as a pde constrained optimization in which the solutions are found using a gradient based descent method recognizing such euler flows as free boundary problems the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients in explicit tracking of interfaces vortex boundaries this method offers an alternative to grid based techniques such as the level set methods and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique we develop and validate this approach using the design of 2d equilibrium euler flows with finite area vortices as a model problem it is also discussed how the proposed methodology can be applied to euler flows featuring other vorticity distributions such as vortex sheets and to time dependent phenomena

a unified coordinates approach to computational fluid dynamics a unified coordinate system is introduced for computational fluid dynamics in which the grid moves with velocity hq q being fluid velocity it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 by suitably choosing the free function h h 1 for 1d flow h chosen to preserve grid angles for 2d flow and to preserve grid skewness for 3d flow the unified coordinate system is shown in a large number of examples involving the euler equations to be superior to both eulerian and lagrangian ones in resolving flow discontinuities shocks and especially slip lines this approach is also successfully extended to shallow water waves and viscous flow

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

an inertia capacitance beam substructure formulation based on the bond graph method with application to rotating beams in this paper a novel inertia capacitance ic beam substructure formulation based on the ic field presentation from the bond graph method is developed the ic beam provides a modular systematic and graphical approach to beam modeling these features allow the modeler to focus more on the modeling and less on the mathematics as such the ic beam is proposed as an alternative to the many existing types of beam models available in the literature the ic beam is formulated in the center of mass body fixed coordinate system allowing for easy interfacing in a multibody system setting this floating frame approach is also computationally cheap elastic deformations in the ic beam are assumed to be small and described by modal superposition the formulation couples rigid body and elastic deformations in a nonlinear fashion the formulation is also compact and efficient detailed derivations for a two dimensional planar ic beam with bending modes are presented a modal acceleration method based on the decoupling of bending modes is proposed for use in the ic beam the rotating beam spin up maneuver problem is solved the karnopp margolis method is applied to ensure complete integral causality for an efficient numerical system geometric substructuring technique is applied to model large deflections the ic beam is shown to be capable of solving the rotating beam problem accurately and efficiently

a two dimensional shear deformable beam element based on the absolute nodal coordinate formulation in this study a new two dimensional shear deformable beam element is proposed for large deformation problems the kinematics of the beam are defined using an exact displacement field where the rotation angles of the cross section caused by bending and shear deformations are described separately cubic interpolation is used for determining the curvature of the beam due to bending while linear interpolation polynomials are used for the shear strain the absolute nodal coordinate formulation in which global displacements and slopes are used as the nodal coordinates is employed for the finite element discretization of the beam the capability of the element to predict static deformation is studied using numerical examples the results imply that the element is free of a phenomenon called shear locking the capability of the element to model highly nonlinear behaviour is established using a bending test where the cantilever is bent into a full circle using only four elements a flexible pendulum and a spin up manoeuvre are modelled in order to study the behaviour of the element in dynamical problems the proposed element is also compared with an existing shear deformable beam element based on the absolute nodal coordinate formulation finally the simple linearization of the beam curvature based on the assumption of small strain will be discussed