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An improvement of rigid bodies contact for particle-based non-smooth walls modeling

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Abstract

Problems of fluid–structure interaction with free-surface flow and multiple bodies are highly nonlinear phenomena, which is challenging for computational modeling and simulation. Particle-based methods have been proven to provide a substantial potential for simulation of free-surface fluid flows and their interactions with multiple solids that are often encountered in fluvial and coastal mechanics fields, due to their Lagrangian tracking scheme and meshless nature. When contact between solids occurs, a numerical modeling to detect it and prevent penetration between bodies is required to avoid numerical inconsistencies. The objective of this work is to propose a solid–solid contact model that relays on contact mechanics theories and, by an effective way to determine the normal direction of contact as well as the distance between bodies faces, eliminates the numerical instabilities induced by non-smooth modeling of bodies surfaces in the particle-based methods. The model adopts a penalty-based method that uses a nonlinear spring and dashpot concept, and thus reproducing the macroscopic properties of the multiple bodies interactions. To avoid very small time step for all the domain, a dynamic sub-cycling algorithm for rigid bodies contact is adopted. First, the present fluid solver, based on the moving particle semi-implicit method, is validated by hydrodynamic and fluid–structure interaction problems. Then, analytical and numerical results are compared, demonstrating the improvements achieved by the proposed model. Finally, the present model is validated via 3D dam breaking events, in which rigid bodies interact with each other or with fixed walls.

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Abbreviations

\( {\mathbf{a}} \) :

Acceleration vector

\( C \) :

Courant number

\( C_{\text{e}} \) :

Effective radius constant

CM:

Center of mass

\( c_{\text{n}} \) :

Normal damping constant

\( c_{\text{t}} \) :

Tangential damping constant

\( d \) :

Number of spatial dimensions

\( {\text{d}}{\mathbf{a}} \) :

Body face area vector

\( E \) :

Young’s modulus

\( {\mathbf{f}} \) :

External body force per unit mass vector

\( {\mathbf{f}}_{\text{c}} \) :

Contact force between rigid bodies vector

\( {\mathbf{f}}^{\text{c}} \) :

Coulomb friction law force vector

\( {\mathbf{f}}^{\text{d}} \) :

Damping force vector

\( {\mathbf{f}}_{\text{ext}} \) :

External force on the rigid body vector

\( {\mathbf{f}}_{\text{g}} \) :

Gravitational force vector

\( {\mathbf{f}}_{\text{h}} \) :

Hydrodynamic force vector

\( {\mathbf{f}}_{\text{n}} \) :

Normal component force vector

\( {\mathbf{f}}^{\text{r}} \) :

Repulsion force vector

\( {\mathbf{f}}_{\text{t}} \) :

Tangential component force vector

\( {\mathbf{g}} \) :

Gravitational acceleration vector

\( g \) :

Absolute value of gravitational acceleration

\( H_{\text{w}} \) :

Water column height

\( i,j \) :

Identification indexes of particles

\( k,l \) :

Identification indexes of the closest particles that are in contact

\( {\mathbf{I}} \) :

Inertia matrix

\( k_{\text{n}} \) :

Normal stiffness constant

\( k_{\text{t}} \) :

Tangential stiffness constant

\( l_{0} \) :

Initial distance between two adjacent particles

\( m \) :

Mass

\( {\mathbf{M}} \) :

Moment vector due to hydrodynamic force

\( {\mathbf{M}}_{\text{c}} \) :

Contact moment vector

\( {\mathbf{M}}_{\text{ext}} \) :

External moment on the rigid body vector

\( {\mathbf{n}}_{\text{c}} \) :

Contact normal vector

\( {\mathbf{n}}_{i} \) :

Normal vector at wall particle i

N :

Particle number density

\( n^{0} \) :

Initial particle number density

\( n^{*} \) :

Particle number density obtained in the explicit part of a time step

N :

Number of particles

\( \hat{N} \) :

Number of particles inside the limited search domain

NB:

Number of bodies

NBp :

Number of bodies in contact with the body p

NC:

Number of pairs of particles belonging to two different bodies that are in contact

P :

Pressure

\( \hat{P} \) :

Minimum pressure of the neighborhood particles

p, q :

Identification indexes of bodies

Q :

Linear momentum

r :

Position vector

\( {\mathbf{r}}_{\text{c}} \) :

Position vector from the center of mass of the rigid body

\( {\mathbf{r}}_{ci} \) :

Position vector between the center of mass of the rigid body and the wall particle i

\( r_{\text{e}} \) :

Effective radius

\( {\mathbf{r}}_{ij} \) :

Distance between two particles i and j

\( {\bar{\mathbf{r}}}_{ij} \) :

Average distance between two particles i and j

\( {\mathbf{r}}_{kl} \) :

Distance between pair of closest particles kl belonging to two different bodies

\( {\mathbf{r}}_{ \bot } \) :

Distance of contact vector

t :

Tangential contact vector

t :

Time

\( u_{\text{w}} \) :

Velocity of wave front

u :

Velocity vector

U :

Total energy

U E :

Elastic energy

\( U_{\text{K}} \) :

Kinetic energy

α :

Artificial compressibility factor

β :

Threshold value of PND for free surface particle detection

γ :

Relaxation coefficient

δ :

Overlap between two wall particles belonging to two different bodies

\( \delta^{t} \) :

Tangential deformation

\( \Delta t_{\text{f}} \) :

Time step used in the fluid domain

\( \Delta t_{\text{s}} \) :

Time step used in the solid domain

λ :

Correction parameter of the Laplacian model of the MPS method

μ :

Kinetic friction coefficient

\( \nu \) :

Poisson’s ratio

\( \nu_{\text{k}} \) :

Kinematic viscosity

\( \xi_{n} \) :

Collision ratio

ρ :

Fluid density

\( \varrho \) :

Weight free surface constant

ϕ :

Scalar function

ω :

Angular velocity vector

ω :

Weight function

Ω w :

Solid wall particles

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Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and by FDTE. The authors are also grateful to Petrobras for financial support on the development of the MPS/TPN-USP simulation system based on MPS method. The authors would also like to thank D.Sc. Ricardo Birjukovs Canelas for the schematic drawing of the experimental configuration.

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  1. Department of Construction Engineering, Polytechnic School of the University of São Paulo, Av. Prof. Almeida Prado, Trav. 2, 83 - Cidade Universitária, São Paulo, SP, 05508-070, Brazil

    Rubens Augusto Amaro Junior, Liang-Yee Cheng & Pedro Henrique Saggioratto Osello

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  1. Rubens Augusto Amaro Junior
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  2. Liang-Yee Cheng
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  3. Pedro Henrique Saggioratto Osello
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Correspondence to Rubens Augusto Amaro Junior.

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Amaro Junior, R.A., Cheng, LY. & Osello, P.H.S. An improvement of rigid bodies contact for particle-based non-smooth walls modeling. Comp. Part. Mech. 6, 561–580 (2019). https://doi.org/10.1007/s40571-019-00233-4

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