A new partitioned staggered scheme for flexible multibody interactions with strong inertial effects

Abstract In generic flexible multibody interaction problems, the differences in the mass, damping and stiffness of the bodies can be very large. Of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure. In this work, a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems. The equation of motion of body with larger inertia is integrated by implicit Newmark method, whereas the smaller inertia is solved using a robust self-starting explicit integration procedure. The coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity, which plays a key role in the stability and accuracy of the simulations. A wide range of mass (10 −6 –10 −1 ), damping and stiffness (10 −3 –10 3 ) ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme. The closed-form expressions for the coupling parameter have been constructed for both matching and non-matching time stepping through the Godunov–Ryabenkii normal mode analysis. The stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system. The time step of the heavier body significantly influences the stability more than the lighter body. To maintain the staggering error minimal, an optimal range of the coupling parameter is identified. The error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods. The partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems. As a potential application of this scheme, flexible multibody problems in the field of offshore engineering, viz., wave energy converter and floater–mooring systems are presented and validated with experimental measurements.

[1]  Fenghua Zhen,et al.  Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method , 2000 .

[2]  Jeffrey W. Banks,et al.  An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids , 2014, J. Comput. Phys..

[3]  Subrata K. Chakrabarti,et al.  Hydrodynamics of Offshore Structures , 1987 .

[4]  Mats Leijon,et al.  Experimental results on power absorption from a wave energy converter at the Lysekil wave energy research site , 2015 .

[5]  A. Dasgupta,et al.  Partitioned viscoplastic-constitutive properties of the Pb-free Sn3.9Ag0.6Cu solder , 2004 .

[6]  Juan J. Alonso,et al.  Multigrid unsteady Navier-Stokes calculations with aeroelastic applications , 1995 .

[7]  Joshua A. White,et al.  Block-partitioned solvers for coupled poromechanics: A unified framework , 2016 .

[8]  M. Leijon,et al.  Theory and Experiment on an Elastically Moored Cylindrical Buoy , 2006, IEEE Journal of Oceanic Engineering.

[9]  D. Peric,et al.  A computational framework for fluid–structure interaction: Finite element formulation and applications , 2006 .

[10]  Tony W. H. Sheu,et al.  Finite element analysis of vortex shedding oscillations from cylinders in the straight channel , 2004 .

[11]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[12]  Jiahuang Tu,et al.  Partitioned subiterative coupling schemes for aeroelasticity using combined interface boundary condition method , 2014 .

[13]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[14]  Jeffrey W. Banks,et al.  An analysis of a new stable partitioned algorithm for FSI problems. Part II: Incompressible flow and structural shells , 2014, J. Comput. Phys..

[15]  K. Tamma,et al.  A robust self‐starting explicit computational methodology for structural dynamic applications: Architecture and representations , 1990 .

[16]  G. Guruswamy.  Unsteady aerodynamic and aeroelastic calculations for wings using Euler equations , 1990 .

[17]  Rajeev K. Jaiman,et al.  Combined interface boundary condition method for coupled thermal simulations , 2008 .

[18]  O. Bauchau.  A self-stabilized algorithm for enforcing constraints in multibody systems , 2003 .

[19]  Scott M. Johnson,et al.  Fully Coupled Geomechanics and Discrete Flow Network Modeling of Hydraulic Fracturing for Geothermal Applications , 2011 .

[20]  Yannis Kallinderis,et al.  Strongly coupled flow/structure interactions with a geometrically conservative ALE scheme on general hybrid meshes , 2006, J. Comput. Phys..

[21]  H. Bijl,et al.  Implicit and explicit higher order time integration schemes for structural dynamics and fluid-structure interaction computations , 2005 .

[22]  Odd Sture Hopperstad,et al.  A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model , 2014 .

[23]  Rajeev K. Jaiman,et al.  Stable and Accurate Loosely-Coupled Scheme for Unsteady Fluid-Structure Interaction , 2007 .

[24]  Miguel Angel Fernández,et al.  Incremental displacement-correction schemes for the explicit coupling of a thin structure with an incompressible fluid , 2011 .

[25]  Oreste S. Bursi,et al.  Convergence analysis of a parallel interfield method for heterogeneous simulations with dynamic substructuring , 2008 .

[26]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[27]  Alberto Cardona,et al.  Energy Preserving Time Integration for Constrained Multibody Systems , 2004 .

[28]  S. K. Godunov,et al.  Theory of difference schemes : an introduction , 1964 .

[29]  Annalisa Quaini,et al.  Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement , 2012, J. Comput. Phys..

[30]  Gang Wang,et al.  Numerical investigation of structural geometric nonlinearity effect in high-aspect-ratio wing using CFD/CSD coupled approach , 2014 .

[31]  Wulf G. Dettmer,et al.  A new staggered scheme for fluid–structure interaction , 2013 .

[32]  Yannis Kallinderis,et al.  Unsteady Flow Structure Interaction for Incompressible Flows Using Deformable Hybrid Grids , 1998 .

[33]  Miguel A. Fernández,et al.  Incremental displacement-correction schemes for incompressible fluid-structure interaction , 2012, Numerische Mathematik.

[34]  E. Kuhl,et al.  An arbitrary Lagrangian Eulerian finite‐element approach for fluid–structure interaction phenomena , 2003 .

[35]  Carlos A. Felippa,et al.  Staggered transient analysis procedures for coupled mechanical systems: Formulation , 1980 .

[36]  Moo-Hyun Kim,et al.  Hull/Mooring Coupled Dynamic Analysis of a Truss Spar In Time Domain , 1999 .

[37]  Hermann G. Matthies,et al.  Algorithms for strong coupling procedures , 2006 .

[38]  Charbel Farhat,et al.  Partitioned procedures for the transient solution of coupled aeroelastic problems , 2001 .

[39]  Frederic Blom,et al.  A monolithical fluid-structure interaction algorithm applied to the piston problem , 1998 .

[40]  D. Tortorelli,et al.  A FETI‐based domain decomposition technique for time‐dependent first‐order systems based on a DAE approach , 2008 .

[41]  Jun Zhang,et al.  Slow motion responses of compliant offshore structures , 1996 .

[42]  V. J. Kurian,et al.  Effect of mooring line configurations on the dynamic responses of truss spar platforms , 2015 .

[43]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[44]  Rajeev K. Jaiman,et al.  Transient fluid–structure interaction with non-matching spatial and temporal discretizations , 2011 .

[45]  Rajeev K. Jaiman,et al.  Combined interface boundary condition method for unsteady fluid–structure interaction , 2011 .

[46]  Suncica Canic,et al.  A partitioned scheme for fluid-composite structure interaction problems , 2015, J. Comput. Phys..

[47]  Ferenc Izsák,et al.  An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model , 2013 .

[48]  Rajeev K. Jaiman,et al.  Fully Coupled Fluid-Structure Interaction for Offshore Applications , 2009 .

[49]  Rajeev K. Jaiman,et al.  A stable second-order scheme for fluid-structure interaction with strong added-mass effects , 2014, J. Comput. Phys..

[50]  T. He.  A Partitioned Implicit Coupling Strategy for Incompressible Flow Past an Oscillating Cylinder , 2015 .

[51]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[52]  van Eh Harald Brummelen,et al.  The relevance of conservation for stability and accuracy of numerical methods for fluid?structure interaction , 2003 .

[53]  Toshiyuki Koto,et al.  IMEX Runge-Kutta schemes for reaction-diffusion equations , 2008 .

[54]  Yan Bao,et al.  Combined interface boundary condition method for fluid–rigid body interaction , 2012 .

[55]  Sunčica Čanić,et al.  Longitudinal displacement in viscoelastic arteries: a novel fluid-structure interaction computational model, and experimental validation. , 2013, Mathematical biosciences and engineering : MBE.

[56]  Charbel Farhat,et al.  Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity , 2006 .

[57]  Michael L. Parks,et al.  Enabling fast, stable and accurate peridynamic computations using multi-time-step integration ☆ , 2016 .

[58]  Miguel Angel Fernández,et al.  Displacement-velocity correction schemes for incompressible fluid-structure interaction , 2011 .

[59]  Charbel Farhat,et al.  Partitioned analysis of coupled mechanical systems , 2001 .

[60]  Dambaru Bhatta,et al.  On scattering and radiation problem for a cylinder in water of finite depth , 2003 .

[61]  Daniele Cavaglieri,et al.  Low-storage implicit/explicit Runge-Kutta schemes for the simulation of stiff high-dimensional ODE systems , 2015, J. Comput. Phys..