Abstract
The main goal of this work is the resolution of fluid structure interaction problems described with the Lagrangian formalism by means of a consistently derived monolithic approach. The use of a component-free derivation leads to a straightforward implementation of the formulation where only vectors and second-order tensors in \({\mathbb {R}}^3\) are required. Therefore, no basis or components have to be imposed ab initio for the discrete variational formulation as occurs when Voigt notation is employed. The computational framework adopted is the local maximum-entropy material point method (LME-MPM), a mesh-free technique that combines the material point sampling of the MPM and the LME, a spatial approximation technique with basis functions of class \(C^{\infty }\). This framework sidesteps the use of expensive mesh refinement techniques, which are typically required when Lagrangian finite element method is employed. Finally, the effectiveness of this approach is illustrated against challenging fluid dynamic problems.
Similar content being viewed by others
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Notes
The ensuing cancellation of linear momentum automatically accounts for dynamic contacts of the seizing type.
To reproduce weakly compressible fluids, the exponent n is set to 7.
It is worth mentioning that although this projector has super-convergence attributes, when it is employed in the context of particle methods this property it is not strictly satisfied since particles are not located, in general, in the quadrature positions.
Otherwise, an extra term must be added to the linearization to represent their variations with respect to \({\varphi }\).
Lagrangian Q4 elements were adopted for the FEM simulation.
Abbreviations
- J :
-
Jacobian of the deformation gradient tensor
- W :
-
Strain energy function
- \({{\nabla _{0}\cdot {\square }}}\) :
-
Divergence operator in the \({\mathcal {B}_0}\) configuration
- \({{\nabla _{0}{\square }}}\) :
-
Gradient operator in the \({\mathcal {B}_0}\) configuration
- \({{\nabla _{n}{\square }}}\) :
-
Gradient operator in the \({\mathcal {B}_n}\) configuration
- \({{\Gamma _0}}\) :
-
Reference boundary
- \({{\Gamma _{\sigma }}}\) :
-
Natural or Neumann boundary conditions over \(\Gamma _0\)
- \({{\Gamma _{\varphi }}}\) :
-
Essential or Dirichlet boundary conditions over \(\Gamma _0\)
- \({{\Psi }}\) :
-
Helmholtz free energy function
- \({{\ddot{\square }}}\) :
-
Second material time derivative
- \({{\dot{\square }}}\) :
-
First material time derivative
- \({{\nabla _{*}{\square }}}\) :
-
Gradient operator which push-forwards the gradient from \({\mathcal {B}_{n}}\) to \({\mathcal {B}_{n+1}}\) during the iterations
- \({{\nabla _{n+1}{\square }}}\) :
-
Gradient operator in the \({\mathcal {B}_{n+1}}\) configuration
- \({{\kappa _f}}\) :
-
Water compressibility
- \({{\textbf{X}}}\) :
-
Material coordinates
- \({{\textbf{x}}}\) :
-
Spatial coordinates
- \({{\mathcal {A}^p}}\) :
-
List of nodes for each particle
- \({{\mathcal {B}_0}}\) :
-
Reference configuration
- \({{\mathcal {B}}}\) :
-
Deformed configuration
- \({{\mathcal {C}_{\varphi }}}\) :
-
Smooth manifold of admissible configurations
- \({{\mathcal {D}^\textrm{int}}}\) :
-
Internal dissipation
- \({{\mathcal {H}^1}}\) :
-
Vector Sobolev space of degree 1
- \({{\mathcal {P}^\textrm{int}}}\) :
-
Stress power
- \({{\mathcal {V}_{\psi }}}\) :
-
Space of the test functions \(\psi \)
- \({{\texttt{B}}}\) :
-
Standard Voigt strain–displacement matrix
- \({{\texttt{D}}}\) :
-
Standard Voigt constitutive matrix
- \({{\mu }}\) :
-
Shear viscosity coefficient
- \({{\nu }}\) :
-
Poisson ratio
- \({{\psi }}\) :
-
Test functions
- \({{\rho }}\) :
-
Describes the scalar density field
- \({{\sigma }}\) :
-
Cauchy stress tensor
- \(\textrm{skew}~({\square })\) :
-
Compute the skew-symmetric part of a tensor
- \({{\square \cdot \square }}\) :
-
Single contraction operator
- \({{\square \circ \square }}\) :
-
Function composition
- \({{\square :\square }}\) :
-
Double contraction operator
- \({{\square \otimes \square }}\) :
-
Dyadic operator
- \({{\square ^I}}\) :
-
Nodal variable
- \({{\square ^p}}\) :
-
Particle variable
- \({{\square ^\textrm{dev}}}\) :
-
Deviatoric component of a tensor
- \({{\square _n}}\) :
-
Time evaluation of a variable at \(t = n\)
- \(\textrm{sym}({\square })\) :
-
Compute the symmetric part of a tensor
- \({{{\textbf{C}}}}\) :
-
Right Cauchy–Green strain tensor
- \({{{\textbf{F}}^*}}\) :
-
Cofactor matrix of the deformation gradient tensor
- \({{{\textbf{F}}^+}}\) :
-
Incremental deformation gradient tensor
- \({{{\textbf{F}}}}\) :
-
Deformation gradient tensor
- \({{{\textbf{I}}}}\) :
-
Identity tensor
- \({{{\textbf{I}}}}\) :
-
Second-order identity tensor
- \({{{\textbf{P}}}}\) :
-
First Piola–Kirchhoff stress tensor
- \({{{\textbf{S}}}}\) :
-
Second Piola–Kirchhoff stress tensor
- \({{\dot{{\textbf{F}}}}}\) :
-
Rate of deformation gradient tensor
- \({{{\mathbf {\tau }}}}\) :
-
Kirchhoff stress tensor \({\tau = J\sigma }\)
- \({{{\textbf{d}}}}\) :
-
Spatial rate of deformation tensor
- \({{{\textbf {a}}}}\) :
-
Acceleration field
- \({{{\textbf {g}}}}\) :
-
Gravity field
- \({{{\textbf {v}}}}\) :
-
Velocity field
- \({{\varphi ^+}}\) :
-
Incremental configuration mapping
- \({{\varphi }}\) :
-
Configuration mapping
- \({{\widehat{\beta }}}\) :
-
Regularization or thermalization parameter of the LME\(_{\beta }\) Pareto set
- m :
-
Mass [M]
- CFD:
-
Computational fluid dynamics
- CFL:
-
The Courant–Friedrich–Levy is defined as, CFL = \({\frac{\text {Cel}}{\Delta x / \Delta t}}\)
- DBC:
-
Dirichlet boundary conditions
- E :
-
Elastic modulus
- Eu:
-
The Euler number shows the dominance of pressure terms over convective terms, Eu = \({\frac{\Delta p}{{\rho }{{\textbf {v}}}^2}}\)
- FEM:
-
Finite element method
- FLIP:
-
Fluid implicit particle
- FSI:
-
Fluid–structure interaction
- IBVP:
-
Initial boundary value problem
- LME:
-
Local maximum entropy
- LME-MPM:
-
Local maximum-entropy material point method
- MPM:
-
Material point method
- NB:
-
Newmark-\(\beta \)
- OTM:
-
Optimal transportation meshfree
- PFEM:
-
Particle finite element method
- PIC:
-
Particle in cell
- PPP:
-
Polynomial pressure projection
- SPH:
-
Smoothed particle hydrodynamics
- TL:
-
Total-Lagrangian
- UL:
-
Updated-Lagrangian
References
Agamloh, E.B., Wallace, A.K., Von Jouanne, A.: Application of fluid-structure interaction simulation of an ocean wave energy extraction device. Renew. Energy 33(4), 748–757 (2008)
Antoci, C., Gallati, M., Sibilla, S.: Numerical simulation of fluid-structure interaction by SPH. Comput. Struct. 85(11–14), 879–890 (2007)
Arroyo, M., Ortiz, M.: Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int. J. Numer. Methods Eng. (2006). https://doi.org/10.1002/nme.1534
Asai, M., Li, Y., Chandra, B., et al.: Fluid-rigid-body interaction simulations and validations using a coupled stabilized ISPH-dem incorporated with the energy-tracking impulse method for multiple-body contacts. Comput. Methods Appl. Mech. Eng. 377(113), 681 (2021)
Balay, S., Abhyankar, S., Adams, M.F., et al.: PETSc/TAO users manual. Tech. Rep. ANL-21/39 - Revision 3.17, Argonne National Laboratory (2022)
Bergamaschi, L., Ferronato, M., Gambolati, G.: Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations. Comput. Methods Appl. Mech. Eng. 196(25), 2647–2656 (2007). https://doi.org/10.1016/j.cma.2007.01.013
Bigoni, D.: Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139178938
Brackbill, J., Kothe, D., Ruppel, H.: Flip: a low-dissipation, particle-in-cell method for fluid flow. Comput. Phys. Commun. 48(1), 25–38 (1988). https://doi.org/10.1016/0010-4655(88)90020-3
Charlton, T.J., Coombs, W.M., Augarde, C.E.: iGIMP: an implicit generalised interpolation material point method for large deformations. Comput. Struct. 190, 108–125 (2017). https://doi.org/10.1016/j.compstruc.2017.05.004
Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13(1), 167–178 (1963). https://doi.org/10.1007/BF01262690
Coombs, W.M., Charlton, T.J., Cortis, M., et al.: Overcoming volumetric locking in material point methods. Comput. Methods Appl. Mech. Eng. 333, 1–21 (2018). https://doi.org/10.1016/j.cma.2018.01.010
Coombs, W.M., Augarde, C.E., Brennan, A.J., et al.: On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity. Comput. Methods Appl. Mech. Eng. 358(112), 622 (2020). https://doi.org/10.1016/j.cma.2019.112622
Cosco, F., Greco, F., Desmet, W., et al.: GPU accelerated initialization of local maximum-entropy meshfree methods for vibrational and acoustic problems. Comput. Methods Appl. Mech. Eng. 366(113), 089 (2020). https://doi.org/10.1016/j.cma.2020.113089
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2009)
de Vaucorbeil, A., Nguyen, V.P., Hutchinson, C.R.: A total-Lagrangian material point method for solid mechanics problems involving large deformations. Comput. Methods Appl. Mech. Eng. 360(112), 783 (2020). https://doi.org/10.1016/j.cma.2019.112783
de Vaucorbeil, A., Nguyen, V.P., Hutchinson, C.R.: aramelo: an open source parallel c++ package for the material point method. Comput. Part. Mech. (2021). https://doi.org/10.1007/s40571-020-00369-8
Degroote, J., Bathe, K.J., Vierendeels, J.: erformance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction. Comput. Struct. 87(11):793–801 (2009). https://doi.org/10.1016/j.compstruc.2008.11.013. Fifth MIT Conference on Computational Fluid and Solid Mechanics
Després, B., Mazeran, C.: Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Ration. Mech. Anal. 178(3), 327–372 (2005). https://doi.org/10.1007/s00205-005-0375-4
Dettmer, W., Peric, D.: A new staggered scheme for fluid-structure interaction. Int. J. Numer. Methods Eng. (2013). https://doi.org/10.1002/nme.4370
Dettmer, W.G., Lovrić, A., Kadapa, C., et al.: New iterative and staggered solution schemes for incompressible fluid-structure interaction based on Dirichlet-Neumann coupling. Int. J. Numer. Methods Eng. 122(19), 5204–5235 (2021)
Doghri, I.: Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects. Springer, Berlin (2013)
Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fluids 46(2), 183–201 (2004)
Fan, J., Liao, H., Ke, R., et al.: A monolithic Lagrangian meshfree scheme for fluid-structure interaction problems within the OTM framework. Comput. Methods Appl. Mech. Eng. 337, 198–219 (2018). https://doi.org/10.1016/j.cma.2018.03.031
Farhat, C., Lesoinne, M.: Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Comput. Methods Appl. Mech. Eng. 182(3–4), 499–515 (2000)
Franci, A., Oñate, E., Carbonell, J.: Unified updated Lagrangian formulation for the analysis of quasi and fully incompressible fluids and solids and their interaction via a partitioned scheme and the PFEM. CIMNE (2014)
Franci, A., Oñate, E., Carbonell, J.M.: Unified Lagrangian formulation for solid and fluid mechanics and FSI problems. Comput. Methods Appl. Mech. Eng. 298, 520–547 (2016). https://doi.org/10.1016/j.cma.2015.09.023
Gilmanov, A., Acharya, S.: A hybrid immersed boundary and material point method for simulating 3D fluid-structure interaction problems. Int. J. Numer. Methods Fluids 56(12), 2151–2177 (2008). https://doi.org/10.1002/fld.1578
Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181(3), 375–389 (1977). https://doi.org/10.1093/mnras/181.3.375
González, D., Cueto, E., Chinesta, F., et al.: A natural element updated Lagrangian strategy for free-surface fluid dynamics. J. Comput. Phys. 223(1), 127–150 (2007). https://doi.org/10.1016/j.jcp.2006.09.002
Greco, F., Filice, L., Peco, C., et al.: A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming. Int. J. Mater. Formg 8, 341–353 (2015). https://doi.org/10.1007/s12289-014-1167-x
Hamad, F.: Formulation of a dynamic material point method and applications to soil-water-geotextile systems. Inst. für Geotechnik (2014)
Hammerquist, C.C., Nairn, J.A.: A new method for material point method particle updates that reduces noise and enhances stability. Comput. Methods Appl. Mech. Eng. 318, 724–738 (2017). https://doi.org/10.1016/j.cma.2017.01.035
Harlow, F.H.: The particle-in-cell method for numerical solution of problems in fluid dynamics (1962)
Hsu, M.C., Bazilevs, Y.: Blood vessel tissue prestress modeling for vascular fluid-structure interaction simulation. Finite Elem. Anal. Des. 47(6), 593–599 (2011). https://doi.org/10.1016/j.finel.2010.12.015
Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola (2000)
Idelsohn, S., Oñate, E., Pin, F.D.: The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int. J. Numer. Methods Eng. 61(7), 964–989 (2004). https://doi.org/10.1002/nme.1096
Idelsohn, S., Marti, J., Limache, A., et al.: Unified lagrangian formulation for elastic solids and incompressible fluids: application to fluid-structure interaction problems via the PFEM. Comput. Methods Appl. Mech. Eng. 197(19), 1762–1776 (2008). https://doi.org/10.1016/j.cma.2007.06.004. Computational Methods in Fluid-Structure Interaction
Kan, L., Zhang, X.: An immersed MMALE material point method for FSI problems with structure fracturing. Comput. Methods Appl. Mech. Eng. 396(115), 099 (2022)
Kane, C., Marsden, J.E., Ortiz, M.: Symplectic-energy-momentum preserving variational integrators. J. Math. Phys. 40, 3353–3371 (1999)
Khayyer, A., Gotoh, H., Falahaty, H., et al.: An enhanced ISPH-SPH coupled method for simulation of incompressible fluid-elastic structure interactions. Comput. Phys. Commun. 232, 139–164 (2018). https://doi.org/10.1016/j.cpc.2018.05.012
Kuhl, E., Steinmann, P.: A hyperelastodynamic ale formulation based on referential, spatial and material settings of continuum mechanics. Acta Mech. 174(3), 201–222 (2005). https://doi.org/10.1007/s00707-004-0200-4
Kularathna, S., Soga, K.: Implicit formulation of material point method for analysis of incompressible materials. Comput. Methods Appl. Mech. Eng. 313, 673–686 (2017). https://doi.org/10.1016/j.cma.2016.10.013
Legay, A., Chessa, J., Belytschko, T.: An Eulerian–Lagrangian method for fluid-structure interaction based on level sets. Comput. Methods Appl. Mech. Eng. 195(17), 2070–2087 (2006) https://doi.org/10.1016/j.cma.2005.02.025. Fluid-Structure Interaction
Li, B., Habbal, F., Ortiz, M.: Optimal transportation meshfree approximation schemes for fluid and plastic flows. Int. J. Numer. Methods Eng. 83(12), 1541–1579 (2010). https://doi.org/10.1002/nme.2869
Li, B., Stalzer, M., Ortiz, M.: A massively parallel implementation of the optimal transportation meshfree method for explicit solid dynamics. Int. J. Numer. Methods Eng. 100(1), 40–61 (2014). https://doi.org/10.1002/nme.4710
Li, M.J., Lian, Y., Zhang, X.: An immersed finite element material point (IFEMP) method for free surface fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 393(114), 809 (2022)
Li, Z., Leduc, J., Nunez-Ramirez, J., et al.: A non-intrusive partitioned approach to couple smoothed particle hydrodynamics and finite element methods for transient fluid-structure interaction problems with large interface motion. Comput. Mech. 55(4), 697–718 (2015). https://doi.org/10.1007/s00466-015-1131-8
Lobovský, L., Botia-Vera, E., Castellana, F., et al.: Experimental investigation of dynamic pressure loads during dam break. J. Fluids Struct. 48, 407–434 (2014). https://doi.org/10.1016/j.jfluidstructs.2014.03.009
Love, E., Sulsky, D.: An unconditionally stable, energy-momentum consistent implementation of the material-point method. Comput. Methods Appl. Mech. Eng. 195(33), 3903–3925 (2006). https://doi.org/10.1016/j.cma.2005.06.027
Macdonald, J.R.: Some simple isothermal equations of state. Rev. Mod. Phys. 38, 669–679 (1966). https://doi.org/10.1103/RevModPhys.38.669
Michler, C., Hulshoff, S., van Brummelen, E., et al.: A monolithic approach to fluid-structure interaction. Comput. Fluids 33(5), 839–848 (2004). https://doi.org/10.1016/j.compfluid.2003.06.006. Applied Mathematics for Industrial Flow Problems
Molinos, M.: The local maximum-entropy material point method. PhD thesis, School of Civil Engineers, Universidad Politécnica de Madrid (2021) https://doi.org/10.20868/UPM.thesis.69327
Molinos, M., Martín Stickle, M., Navas, P., et al.: Towards a local maximum-entropy material point method at finite strain within a b-free approach. Int. J. Numer. Methods Eng. 122, 5594–5625 (2021). https://doi.org/10.1002/nme.6765
Molinos, M., Navas, P., Pastor, M., et al.: On the dynamic assessment of the Local-Maximum Entropy Material Point Method through an Explicit Predictor-Corrector Scheme. Comput. Methods Appl. Mech. Eng. 374(113), 512 (2021). https://doi.org/10.1016/j.cma.2020.113512
Molteni, D., Colagrossi, A.: A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Comput. Phys. Commun. 180(6), 861–872 (2009). https://doi.org/10.1016/j.cpc.2008.12.004
Monaghan, J.: Simulating free surface flows with SPH. J. Comput. Phys. 110(2), 399–406 (1994). https://doi.org/10.1006/jcph.1994.1034
Morikawa, D.S., Asai, M.: Coupling total Lagrangian SPH-EISPH for fluid-structure interaction with large deformed hyperelastic solid bodies. Comput. Methods Appl. Mech. Eng. 381(113), 832 (2021). https://doi.org/10.1016/j.cma.2021.113832
Murakami, S., Ha, B.H., NAKAO, H., et al.: Sph analysis on tsunami flow around bridge girder. J. Jpn. Soc. Civil Eng. Ser A1 (Struct. Eng. Earthq. Eng. (SE/EE)) 65(1), 914–920 (2009). https://doi.org/10.2208/jscejseee.65.914
Navas, P., López-Querol, S., Yu, R.C., et al.: Optimal transportation meshfree method in geotechnical engineering problems under large deformation regime. Int. J. Numer. Methods Eng. (2018). https://doi.org/10.1002/nme.5841
Newmark, N.M.: A method of computation for structural dynamics. J .Eng. Mech. Div. 85, 67–94 (1959)
NL-PartSol. Non-linear Particle Solver (2022). https://github.com/migmolper/NL-PartSol
Oñate, E., Carbonell, J.M.: Updated Lagrangian mixed finite element formulation for quasi and fully incompressible fluids. Comput. Mech. 54(6), 1583–1596 (2014). https://doi.org/10.1007/s00466-014-1078-1
Planas, J., Romero, I., Sancho, J.: B free. Comput. Methods Appl. Mech. Eng. 217–220, 226–235 (2012). https://doi.org/10.1016/j.cma.2012.01.019
Radovitzky, R., Ortiz, M.: Lagrangian finite element analysis of Newtonian fluid flows. Int. J. Numer. Methods Eng. 43(4), 607–619 (1998). https://doi.org/10.1002/(SICI)1097-0207(19981030)43:4<607::AID-NME399>3.0.CO;2-N
Riemann, B.: Über die darstellbarkeit einer function durch eine trigonometrische reihe. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13, 87–138 (1854). https://doi.org/10.1016/j.cma.2017.09.022
Rosolen, A., Arroyo, M.: Blending isogeometric analysis and local maximum entropy meshfree approximants. Comput. Methods Appl. Mech. Eng. 264, 95–107 (2013). https://doi.org/10.1016/j.cma.2013.05.015
Ryzhakov, P.B., Rossi, R., Idelsohn, S.R., et al.: A monolithic Lagrangian approach for fluid-structure interaction problems. Comput. Mech. 46(6), 883–899 (2010). https://doi.org/10.1007/s00466-010-0522-0
Simo, J., Tarnow, N., Wong, K.: Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Methods Appl. Mech. Eng. 100(1), 63–116 (1992). https://doi.org/10.1016/0045-7825(92)90115-Z
Simo, J.C., Tarnow, N.: The discrete energy-momentum method. conserving algorithms for nonlinear elastodynamics. Zeitschrift für angewandte Mathematik und Physik ZAMP 43 (1992). https://doi.org/10.1007/BF00913408
Song, Y., Liu, Y., Zhang, X.: A non-penetration FEM-MPM contact algorithm for complex fluid-structure interaction problems. Comput. Fluids 213(104), 749 (2020)
Steffen, M., Kirby, R.M., Berzins, M.: Analysis and reduction of quadrature errors in the material point method (MPM). Int. J. Numer. Methods Eng. 76(6), 922–948 (2008). https://doi.org/10.1002/nme.2360
Stein, K.R., Tezduyar, T.E., Kumar, V., et al.: Numerical simulation of soft landing for clusters of cargo parachutes. Adv. Appl. Mech. 28, 1–44 (1992)
Stickle, M.M., Molinos, M., Navas, P., et al.: A component-free Lagrangian finite element formulation for large strain elastodynamics. Comput. Mech. (2022). https://doi.org/10.1007/s00466-021-02107-0
Sulsky, D., Kaul, A.: Implicit dynamics in the material-point method. Comput. Methods Appl. Mech. Eng. 193(12), 1137–1170 (2004). https://doi.org/10.1016/j.cma.2003.12.011
Sulsky, D.L., Schreyer, H., Chen, Z.: A particle method for history-dependent materials. Comput. Methods Appl. Mech. Eng. 118(1), 179–196 (1994). https://doi.org/10.1016/0045-7825(94)90112-0
Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells. Springer, Berlin (1959)
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1999). https://doi.org/10.1007/978-3-662-03915-1
Vilar, F., Maire, P.H., Abgrall, R.: A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids. J. Comput. Phys. 276, 188–234 (2014). https://doi.org/10.1016/j.jcp.2014.07.030
Walhorn, E., Kölke, A., Hübner, B., et al.: Fluid-structure coupling within a monolithic model involving free surface flows. Comput. Struct. 83(25), 2100–2111 (2005). https://doi.org/10.1016/j.compstruc.2005.03.010
Willcox, K., Paduano, J., Peraire, J., et al.: Low order aerodynamic models for aeroelastic control of turbomachines. In: 40th Structures, Structural Dynamics, and Materials Conference and Exhibit, p. 1467 (1999)
Wriggers, P.: Nonlinear Finite Element Methods, 1st edn. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-71001-1
Wriggers, P., Simo, J.C.: A general procedure for the direct computation of turning and bifurcation points. Int. J. Numer. Methods Eng. 30(1), 155–176 (1990). https://doi.org/10.1002/nme.1620300110
York, A.R., II., Sulsky, D., Schreyer, H.L.: Fluid-membrane interaction based on the material point method. Int. J. Numer. Methods Engi. 48(6), 901–924 (2000). https://doi.org/10.1002/(SICI)1097-0207(20000630)48:6<901::AID-NME910>3.0.CO;2-T
Zhang, C., Rezavand, M., Hu, X.: A multi-resolution SPH method for fluid-structure interactions. J. Comput. Phys. 429(110), 028 (2021). https://doi.org/10.1016/j.jcp.2020.110028
Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24(2), 337–357 (1987). https://doi.org/10.1002/NME.1620240206
Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int. J. Numer. Methods Eng. 33(7), 1331–1364 (1992). https://doi.org/10.1002/nme.1620330702
Acknowledgements
This research was funded by the Ministerio de Ciencia e Innovación, under Grant Number, PID2019-105630GB-I00. M. Molinos appreciates the Fundación Entrecanales Ibarra for his PhD fellowship and thanks the Universidad Politécnica de Madrid for the financial support for his research stay in the University of California Berkeley. B. Chandra gratefully acknowledges the support from the Jane Lewis Fellowship of the University of California, Berkeley. Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no potential conflict of interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A Auxiliary derivations
Lemma 1
Let \(\delta {\varphi }\) be the variation of the differentiable mapping \({\varphi }\); then, the linearization of the deformation gradient \({{\textbf{F}}}\) is given by:
Proof of Lemma 1
The linearization of the deformation gradient has the following structure:
\(\square \)
Lemma 2
The linearization of \({{\textbf{F}}}^{-1}\) reads as:
Remark
An immediate result of Lemma 2 is:
Proof of Lemma 2
Consider the relation \({\textbf{A}}^{-1}{\textbf{A}} = {{\textbf{I}}}\) as starting point:
Therefore, the variation of the inverse for any non-singular tensor \({\textbf{A}}\) can be expressed as Eq. (A5). \(\square \)
Lemma 3
Let J be the determinant, or Jacobian, of the deformation gradient \({{\textbf{F}}}\) and \({{\textbf{F}}^*}\) the cofactor of \({{\textbf{F}}}\). Then, the linearization of J is:
Proof of Lemma 3
For a \({\mathbb {R}}^3 \times {\mathbb {R}}^3\) non-singular matrix \({\textbf{A}}\), i.e., one with nonzero determinant, the Cayley–Hamilton theorem states the following characteristic equation:
\(I_{2}\) is the second invariant of \({\textbf{A}}\). For the particular case of \({{\textbf{F}}}\) and rearranging the expression of the characteristic equation:
Considering the result obtained in Lemma 1:
\(\square \)
Lemma 4
The linearization of the cofactor of the deformation gradient \({{\textbf{F}}}\) is:
Proof of Lemma 4
Following the definition of \({{\textbf{F}}^*}\), applying the chain rule over it and considering the intermediate results in Lemmas 1 and 3:
\(\square \)
Lemma 5
The linearization of the rate of the deformation gradient \({\dot{{\textbf{F}}}}\) is:
Proof of Lemma 5
Taking as starting point the linearization of the deformation gradient obtained in Lemma 1 and having in mind the time discretization presented in Sect. 3.2:
\(\square \)
Lemma 6
The linearization of the material velocity gradient \(\nabla {{{\textbf {v}}}}\) is:
Remark
By the application of Lemma 6, the linearization of \({{\textbf{d}}}\) is:
Proof of Lemma 6
Considering the definition of \(\nabla {{{\textbf {v}}}}\) as in Eq. (5) and applying the intermediate results obtained in Lemma 2 and Lemma 5:
\(\square \)
Lemma 7
For any vectors \({{\textbf {v}}},{{\textbf {b}}},\delta {{\textbf {a}}} \in {\mathbb {R}}^{dim}\) and any fourth-order tensor \({\mathbb {T}}: \square \), the following relation is satisfied:
This leads to the definition of the operator:
From this definition, it is a simple exercise to find the bilinear tensor form \({\mathbb {T}} \{\square ,\square \}\) for the various simple, but frequent, tensor-valued linear operations collected in Table 7.
Remark
Consider as example of application \({{\textbf {v}}} = \nabla _{0}{N({{\textbf {x}}})}^{\beta }\), \({{\textbf {b}}} = \nabla _{0}{N({{\textbf {x}}})}^{\alpha }\), \(\delta {{\textbf {a}}} = D{\varphi }_{n+1}^{\beta }\), and \({\mathbb {T}}: \square = {\mathbb {A}}: \square \). We pretend to factorize out the variation of the displacement, \(D{\varphi }^{\beta }_{n+1}\), in the foregoing expression:
By the application of Lemma 7:
Proof of Lemma 7
See Planas et al. [63]. \(\square \)
Appendix B Linearization of the variational equation for the solid phase
The component-free expression of the tangent density \({\mathbb {A}} \big \{\square ,\square \big \}\) for a general isotropic hyperelastic material whose strain energy density function W is function of the principal invariants of \({{\textbf{C}}}\) is given by Stickle et al. [73]:
where
The coefficients \(\Gamma _i\) are scalar functions of the principal invariants of \({{\textbf{C}}}\), see Doghri [21]. The full development of Eq. (B17) has been omitted for the sake of clarity. Interested readers are recommended to refer to Stickle et al. [73] for the detailed derivation. For the particular case of a Neo-Hookean material, these coefficients \(\Gamma _i\) are:
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Molinos, M., Chandra, B., Stickle, M.M. et al. On the derivation of a component-free scheme for Lagrangian fluid–structure interaction problems. Acta Mech 234, 1777–1809 (2023). https://doi.org/10.1007/s00707-022-03459-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-022-03459-1