Coupling Analysis of FEM for Free-Surface Fluids and Rigid Body
Kanji Fukushima1, Seizo Tanaka2 and Daigoro Isobe31

Graduate School, University of Tsukuba
Ibaraki 305-8573, Japans1630201@s.tsukuba.ac.jp

2

Division of Engineering Mechanics and Energy, University of Tsukuba
Ibaraki 305-8573, Japan
stanaka@kz.tsukuba.ac.jp
3
Division of Engineering Mechanics and Energy, University of Tsukuba
Ibaraki 305-8573, Japan
isobe@kz.tsukuba.ac.jp
This paper presents a coupling method of rigid body motion and fluid dynamics analysis. Thismethod will be required when we develop a numerical code for damage evaluation ofstructures under collision of tsunami debris. A debris is assumed to be a simple-shaped rigid
body and a process of calculation of interaction between the rigid body and fluid FEM isdescribed. Furthermore, the results are validated by a test result.
As for the fluid analysis method, the SUPG/PSPG stabilized FEM [1] and an interface
capturing approach based on VOF technique [2] are adopted. There are three reasons that wechoose FEM for fluids. The first is that FEM can express complicated domain shapes and canobtain accurate solutions by changing the resolution of mesh. It reduces excessive
computational cost. The second is that it can easily achieve better efficiency in parallel
computing by using a domain decomposition method. The adopted method is based on a
stationary mesh and does not require any dynamic load balancing techniques as used inparticle method [3]. The last one is that it is easier to calculate the physical values in domain.
FEM can obtain the distribution of the physical values as well as the values at any point indomain with an element piece-wise function. They are used to calculate the interactive force
between fluid and rigid body.
Fluid force applied to a rigid body is calculated by integrating fluid stress tensors on the rigid
body’s surface area included in each fluid element. The surface area is considered to be flat in
the fluid elements, which lead us to calculate on the area of two types of shapes, triangle andquadrilateral. The surface integral of these values is calculated using shape functions of aplane element, and fluid force vectors are obtained. The force vectors are then applied to rigid
body analysis. On the other hand, velocity and angular velocity are obtained by the rigid bodyanalysis. The fluid velocity at the nodes overlapped with rigid body can be geometricallycalculated, and it is applied to fluid analysis as constraint condition.
The proposed method is verified by simulating a rigid sphere motionlessly floating on thewater surface. Figure 1 shows the overview of the numerical model. The diameter of thesphere is 20 mm and the density is half of the water. Figure 2 shows the displacements in Xand Z directions divided by the diameter, obtained by the analysis. Small vibration can beobserved in Z-direction, however, the sphere remains stable until 1.5 s in X-direction. Then, itstarts to move largely in X-direction, which is assumed to be caused by numerical oscillation.

airrigid sphere

water(a) X-direction

ZYX

Fig.1 Numerical model ofa floating rigid sphere(b) Z-direction
Fig.2 Displacement divided by diameter of sphere

airrigid cylinder
free fallwater(a) 0.4 sFig.3 Test subject for a rigid cylinder
free-falling in water(b) 0.55 s(c) 0.65 s

Fig.4 Contours of vorticity obtained bynumerical analysis

Furthermore, the developed method is validated by comparing the numerical result with test
results [4]. Figure 3 shows the overview of the test. Figure 4 shows the contours of vorticity
obtained by the analysis at the time when the trailing vortex occurs. It is confirmed from thenumerical results that the interaction between fluid and rigid body works practically well,although the computation process of the fluid force should still be improved.
References[1] T. E. Tezduyar, Stabilized finite element formulations for incompressible flow
computations, Advances in applied mechanics, 28, pp. 1-44, 1992.
[2] S. Aliabadi, T. E. Tezduyar, Stabilized-finite-element/interface-capturing technique forparallel computation of unsteady flows with interfaces, Comput. Methods Appl. Mech. Engrg.,
190, pp. 243-261, 2000
[3] M. Furuichi, S. Tsuzuki and H. Sakaguchi, Computation-communication overlap
techniques for the particle method implemented a dynamic load balancing, JSME 29th
Computational Mechanics Division Conference, 230, September 2016, in Japanese.
[4] S. Ushijima et al., Computational Method for Multiphase Incompressible Flows (MICS)
and its Applicability to Particle-Laden Liquid Flows, Journal of JSCE, No. 740/II-64, pp.
121-130, August 2003, in Japanese.