Design of a computational model for multi-body contact of deformable bodies in the flowing fluid

The article deals with the design and testing of a computational model for multi-body contact of deformable bodies in the flowing fluid. The computational model will be designed to allow easy modify the impact area of deformable bodies and their shape. The computational model must allow the application of an endless cycle of bodies impact with the possibility of restarting and calculating the following time intervals for collision of bodies.


Modelling of multi-body contact in the fluid flow
Modelling of a multi-body contact is a computationally demanding task. The computational difficulty of a model increases with a number of bodies that may come into contact. If we consider a possible contact among all bodies in the solved task, then a number of contact pairs will be determined according to the relation P = (n-1) 2 , where n is the number of all bodies in the contact problem. When generating a mesh of finite elements, it is necessary to define a sufficiently dense mesh of finite elements on the contact bodies. The contact task has bad convergence in case of low mesh density. Mesh density must be also determined with regard to shape complexity of bodies in such a way it approximates an edge of a contact body with sufficient precision. Most of computational models of the contact implemented in software for computing by the method of finite elements divide contact bodies into a group of 'contactor' and 'target'. In such a case the contact is detected in case when nodal points of the body 'contactor' penetrate into a body of the type 'target'. On the contrary, if nodal points of the mesh of a body 'target' penetrate into a body of 'contactor', a contact is not detected. In case of a multi-body contact it is convenient to define contact pairs in such a way that each body is a 'contactor' and at the same time also a 'target'. By such a definition of contact pairs convergence of a solution of the contact task becomes rapidly improved and oscillation in nodal points is decreased [1][2][3].
Computational models for solving of flowing fluids require a fine mesh of elements, elements must be sufficiently small so that a flow in the boundary layer close to a body surface would be possible to solve. Both groups of computational models, contact tasks and flow, belong to a group of models with a high demand on computational means and time [4][5][6][7][8][9].
The aim of the design of a computational model is to design an efficient computational model for the analysis of a mutual contact of bodies in flowing fluid [10][11]. The designed computational model must contain following properties: -A part of the model must be constructed in a way to enable mutual collisions of contact bodies and a shape of the area must be easily modified. -A channel in which fluid flows and carries contact bodies must be closed, bodies proceed through the channel and re-collide. -A computational model must be maximally efficient from the standpoint of requirements on computational means and computing time. -A designed computational model will be created for the software package ADINA, a designed model will consider possibilities of a model design in this program. -Contact bodies will have a circular shape.
-A shape of the channel must be designed to avoid mutual jamming and a blockage of the channel. -A task will be solved in 2D while structural bodies will have plane state of stress.
-In flowing, a material model for viscous flow regardless a turbulence model will be considered.

The geometric and FE model for the structural analysis
The designed geometric model of an enclosed channel in the structural analysis is formed by the following elements. The geometric model is shown in Figure 1: -A structure forming an outer boundary of the channel. Geometry of the body is designed in the system ADINA via an application of the module ADINA-M which is the module for modelling of bodies by the use of a library for a graphic format parasolid. A gradual shape of the channel is designed in a way to prevent mutual jamming of contact bodies and the channel blockage.  [12][13]. A diameter of contact bodies is 10 mm, the mean diameter of the circle in curves of the channel is 60 mm and the width of the channel is 11 mm.

Fig. 1. Geometry of structural model and boundary condition
The program ADINA enables a definition of several contact groups, in a contact group contact bodies and contact pairs are defined. Due to less complex modelling, a design of a model, we defined following contact groups: -The first contact group -a mutual contact of contact bodies. -The fifth contact group -a contact of 46 circular contact bodies and 18 contact bodies of auxiliary geometry. There are 828 contact pairs in this group. In the first four contact groups we must define an occupation of contact areas within small distances and we use the value 5x10 -2 mm in the model while the diameter of contact bodies is 10 mm. Occupation of a body contact is necessary for a correct function of the model of flow. Among contact bodies there must be formed a thin layer of flowing elements -at least one element. Otherwise, it would not be possible to generate a mesh of finite elements for a model of the flowing fluid, boundaries of an area in the model of flowing must not overlap or contact.

-A zero initial speed is defined for all bodies.
A calculation is solved as an implicit task by Newmark method, for plane stress statea thickness of designed geometric model is 100 mm. Thickness of 100 mm is selected thus maximum contact forces will not exceed 100 N. Properties of the contact are selected in a way that contact forces lower than 0.1 N are not considered in the contact. Limits of a minimum value of contact forces strikingly improve convergence of a contact task and it prevents oscillation in contact detection.
Due to a reason of the maximum efficiency of a computing model we selected a mesh density to generate a minimum number of elements in which a convergence of the task is secured. In this case values of contact forces are calculated correctly, but the state of stress in the contact point is not. In a case we would intend to determine the state of stress with sufficient precision in the contact point, i tis necessary to increase density and, eventually, to change a type of elements into quadratic elements. In generating of a mesh on contact bodies we use 10 elements per a quarter of a circle. In the other parts of the model, the size of elements is set on 2 mm. Auxiliary lines are modelled via an application of one beam element per a line. The final part of the auxiliary lines is modelled with a small cross section of the beam and beam elements form an elastic element in this case. A presence of an elastic element in the model decreases maximum values of contact forces among circular contact bodies and auxiliary lines in an application of greater time span [14]. A mesh of linear triangular elements for a structural model is depicted in Figure 2.

The geometric and FE model for analysis of fluid flow
When designing a geometric model of flow we follow geometry designed for the structural model. For designing an outer boundary of the area boundary curves from the structural model are applied and they are supplemented by the inflow and outflow parts of the model. Boundaries of contact bodies are defined by a construction of a circle with a rotating point of 360°. Consequently, bodies in the module of ADINA-M are designed from these boundary curves and the geometric model for a boundary in the area of fluid in the model is formed by their mutual subtracting. The geometric model is depicted in Figure 3.
In the model there is applied a model of a laminar flow and following properties of the fluid (water) are defined: dynamic viscosity 0.001 kgm -1 s -1 , fluid density 1000 kg m -3 .
In the designed model of flow there are defined following boundary and initial conditions: -The boundary of contact bodies is defined as "no slip FSI boundary". When designing a mesh of finite elements the FCBI-C formulation of elements for flow is applied. In FCBI-C formulation of finite elements is defined one control point in the centre of gravity in every element. FCBI-C formulation enables an application of a segregated iteration solver AMG. A selection of a formulation of elements and a type of a solver is designated with a regard of minimisation of a time period needed for a solution. In meshing we used triangle elements of the size 0.2 mm and in the process of solution the automatic meshing of finite elements for flow is activated.
In solving, the constant time step 1x10 -4 s is applied.  In numerical tests of the designed computational model, correctness of a design of a gradual constriction of the channel was proved. In this spot the mutual jamming of circular contact bodies does not take place.
Presented results of contact forces are shown in the time of 0.2711 in Figure 4. The speed of fluid flow is shown in Figure 5 in the same time; the speed of flow is shown in a form of velocity vectors ms -1 .

Conclusion
The designed FE computational model meets all required criteria imposed on the model. It enables a mutual interaction of contact bodies together with an interaction with channel boundaries. The interaction of bodies takes place in the flowing fluid, while the speed of fluid flow is possible to modify regardless the speed of rotation of auxiliary lines formed by beam elements.
In testing calculations was proved that a gradual channel shape is correctly designed and it enables gradual ordering of bodies into a taper part of the channel. In a transition of about 50 bodies their mutual jamming did not occur.