A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures

TLDR: In this article, a sheet of paper is modeled as a curve by the dynamical Kirchhoff-love theory and the external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet.

Abstract: We present the application of a meshfree method for simulations of interaction between fluids and flexible structures As a flexible structure we consider a sheet of paper In a two-dimensional framework this sheet can be modeled as curve by the dynamical Kirchhoff-Love theory The external forces taken into account are gravitation and the pressure difference between upper and lower surface of the sheet This pressure difference is computed using the Finite Pointset Method (FPM) for the incompressible Navier-Stokes equations FPM is a meshfree, Lagrangian particle method The dynamics of the sheet are computed by a finite difference method We show the suitability of the meshfree method for simulations of fluid-structure interaction in several applications

Figures

Figure 6. Position of particles and sheets of paper (top), pressure distribution (center) and velocity field (bottom) at t = 0.15 s

Table 1.

Figure 2. Position of particles and sheet of paper at time t = 0 s and t = 0.8377 s (top), pressure and velocity field at t = 0.8377 s (bottom)

Figure 5. Computational domain

Figure 1. Orientations of the particles around a sheet of paper, blue indicates orientation 0, red +1 and yellow −1

Figure 4. Velocity field at t = 0.0, 0.3 s (top), t = 0.6, 0.9 s (center) and t = 1.2, 1.5 s (bottom)

Full text

S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, R. Wegener
A Meshfree Method for Simulations
of Interactions between Fluids and
Flexible Structures
Berichte des Fraunhofer ITWM, Nr. 88 (2006)

© Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM 2006
ISSN 1434-9973
Bericht 88 (2006)
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A Meshfree Method for Simulations of
Interactions between Fluids and Flexible
Structures
Sudarshan Tiwari, Sergey Antonov, Dietmar Hietel, J¨org Kuhnert,
Ferdinand Olawsky and Raimund Wegener
Fraunhofer-Institut f¨ur Techno- und Wirtschaftmathematik (ITWM)
Gottlieb-Daimler-Straße, Geb. 49, D-67663 Kaiserslautern, Germany
tiwari@itwm.fraunhofer.de
Summary. We present the application of a meshfree method for simulations of
interaction between fluids and flexible structures. As a flexible structure we consider
a sheet of paper. In a two-dimensional framework this sheet can be modeled as curve
by the dynamical Kirchhoff-Love theory. The external forces taken into account are
gravitation and the pressure difference between upper and lower surface of the sheet.
This pressure difference is computed using the Finite Pointset Method (FPM) for
the incompressible Navier-Stokes equations. FPM is a meshfree, Lagrangian particle
method. The dynamics of the sheet are computed by a finite difference method.
We show the suitability of the meshfree method for simulations of fluid-structure
interaction in several applications.
Key words: Meshfree Method, FPM, Fluid Structure Interaction, Sheet of
Paper, Dynamical Coupling
1 Introduct ion
There exists a wide range of problems where the motion of flexible structures
is driven by a fluid flow. Such kind of problems are in printing processes the
prediction of the fluttering of a thin sheet [11] or in bio-mechanics the opening
and closing behavior of aortic heart valves [2]. In general, the structure is
considered as a moving wall and its dynamics are described in a Lagrangian
framework. Therefore, it is suitable to use a Lagrangian formulation also for
the fluid. Whenever the structure moves rapidly, mesh based methods like
finite volume or finite element methods are limited due to grid adaption by re-
meshing during the simulation. It requires the use of interpolation techniques
to recover the fluid dynamical variables on the new mesh. This procedure not
only introduces artificial diffusivity, but is also problematic with respect to

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A meshfree method for simulations of interactions between fluids and flexible structures" ?

In this paper, the Finite Pointset Method ( FPM ) is used to handle a wide range of dynamical fluid structure interactions in a two-dimensional framework that couples FPM with the dynamics of a onedimensional sheet. 

Q2. What future works have the authors mentioned in the paper "A meshfree method for simulations of interactions between fluids and flexible structures" ?

Future work will be the extension of the method to three dimensional problems. 

Q3. What is the way to simulate the dynamical interaction between fluid and structure?

Since the sheet of paper has very small bending stiffness and low weight, the grid size as well as the time step for the simulation of its dynamics should be very small. 

Q4. What is the simplest way to solve the Navier-Stokes equations?

Use the new positions and velocities of boundary particles of the prior step to solve the Navier-Stokes equations and obtain new positions xn+1 and flow fields vn+1 and pn+1(iv) 

Q5. How many particles should be on the same line?

To ensure consistency in two dimensions the number m of these particles should be at least 6 and they should neither be on the same line nor on the same circle. 

Q6. What is the orientation of the boundary particle?

Then all inner particles are marked with the orientation +1 if they are in the neighborhood of a boundary particle of orientation +1 and if they lie on the half plane defined by the normal of this boundary particle. 

Q7. What is the simplest way to handle the sheet of paper?

Either an end is fixed with given position and direction byr(0, t) = r0(t), ∂sr(0, t) = e0(t) (6)or it is free with Neumann boundary conditions given by∂ssr(L, t) = 0, ∂sssr(L, t) = 0, T (L, t) = 0. (7)In some applications the sheet of paper can hit a wall. 

Q8. What is the resulting system for the unknowns a?

The resulting system for the unknowns a = (ψx, ψy, ψxx, ψxy, ψyy)T is not under-determined for m > 5 but can in general only be solved as a least squares solution by minimizing the error e in the resulting systeme = 

Q9. What is the second method presented in [9]?

It is found that the method presented in [9] is more stable and that Neumann conditions can be easily included in the approximation. 

Q10. What is the problem with the meshfree method?

This procedure not only introduces artificial diffusivity, but is also problematic with respect toaccuracy, robustness and efficiency. 

Q11. What is the resulting system for inner grid points i?

The resulting system for inner grid points i = 1, · · · , N − 1 is given byω rn+1i − 2 rni − rn−1iΔt2 = 1 Δs[ T n+1i+ 12 (∂sr)n+1i+ 12 − T n+1 i− 12 (∂sr)n+1i− 12] −BΔs [ (∂sssr)n+1i+ 12 − (∂sssr)n+1i− 12 ] + [p]ni n n i + ω g. (12)Here, the normal vector ni is orthogonal to the tangential vector defined by the average of (∂sr)i± 12 . 

Q12. What is the bending stiffness of a sheet of paper?

The dynamics of the sheet of paper is given by balancing the acting forces with the accelerationω ∂ttr(s, t) = ∂s[T (s, t)∂sr(s, t)] − B ∂ssssr(s, t) + [p] n + ω g. (4)Here, the unknown tension T acts as a Lagrangian multiplier referencing to the additional constraint of in-extensibility‖∂sr(s, t)‖2 = 1. (5)The base weight ω and the bending stiffness B are given material constants of the sheet of paper. 

Q13. How many neighbors are allowed in the iterative solvers?

in order to obtain the accuracy of the weighted least squares approximation the scheme guarantees everywhere a minimum number of neighbors. 

Q14. What is the solution of the minimization problem?

The solution now minimizes the functionalJ̃ = m+2∑ i=1 wie 2 i = (M̃ ã − b̃)T W̃ (M̃ ã − b̃) (27)with W̃ = diag (w1, . . . , wm, 1, 1) and wi = w(xi −x; h) for i = 1, . . . , m while wm+1 = wm+2 = 1. Formally, the solution of this minimization problem can be written asã = (M̃T W̃M̃)−1(M̃T W̃ )b̃. (28)The value of ψ is the only relevant component of the solution ã. 

Q15. What is the velocity field in the first row?

In the second row the authors have plotted the pressure (left), with values varying from 0.29512 Pa (blue) to 0.84271 Pa (red) and the velocity field (right) with maximum value 0.059941 m/s (red) at time t = 0.8377 s.