### Tech Briefs

**Simulation of a fluid-filled rubber capsule compressed by a cylindrical rod**

ADINA is known for its powerful capabilities in the solution of fluid structure interaction (FSI) problems. For many years, ADINA FSI solutions have been used successfully for many applications in a wide range of industries, see ADINA FSI.

Although ADINA FSI is a powerful tool for many problems involving structures and fluid flow, there are certain classes of problems involving structures and fluids that do not require the use of ADINA FSI but can be solved effectively through a purely structural analysis. An example of such problems is the analysis of fluid-filled cavities inside a solid or structure. In this Tech Brief, we demonstrate the use of the hydrostatic fluid feature in ADINA Structures that can be used to model the pressure and volume variation in cavities filled with compressible fluids. An example of such application is the simulation of a compressed fluid-filled rubber capsule as shown in the animation above.

For the hydrostatic fluid model, it is assumed that the fluid pressure in the cavity is uniform at any point of time. The model only requires the hydrostatic fluid pressure to be applied on the internal surface of the cavity, with no material definition required. The variation of the hydrostatic fluid pressure and the volume of the cavity is governed by the ideal gas law below.

By default, the ambient pressure is zero which corresponds to the ideal situation when there is no environmental pressure. In the real world however, the ambient pressure is usually nonzero, and in most case it is the atmospheric pressure. Although a nonzero ambient pressure does not directly act on the model, it can dramatically affect the value of the gauge pressure.

The example below illustrates a benchmark solution of the hydrostatic pressure inside a rectangular space enclosed by six surrounding faces as shown in Figure 1. The top and bottom blocks are meshed with solid elements while the four vertical sides are dummy segments without elements used to form the enclosed volume. Hydrostatic pressure is applied on all six inside faces of the enclosed space. A prescribed displacement is applied on the top block to compress the enclosed space to the half of its original volume.

**Figure 1**

In the first stage of the simulation, time *t* = 0 to 1, the initial pressure is gradually increased from 0 to 1000; then in the second stage, *t* = 1 to 2, the space is compressed to half the original size. Two different cases, with and without ambient pressure, are separately considered. In the case with non-zero ambient pressure, the ambient pressure also takes the value 1000. The variation of the calculated hydrostatic fluid pressure (gauge pressure) with time for both cases is shown in the video of Figure 2 below. The calculated solutions correspond exactly to the analytical solutions.

**Figure 2**

In a hydrostatic fluid pressure analysis, the model can contain multiple cavities as illustrated in the 2D example of Figure 3 below. The left part of the model is a rectangular region with the top and bottom meshed with 2D elements and the two side edges modeled with dummy segments, similar to the 3D model above. The right part of the model contains an irregular cavity enclosed by a relatively soft material meshed with 2D elements. When the left cavity is compressed, the hydrostatic pressure in both enclosed cavities increases, resulting in the expansion of the cavity on the right. By adjusting the time functions assigned to the initial pressure and the prescribed displacement, the repeated compression and decompression of the left cavity can be used to simulate a pumping process.

**Figure 3**

Another interesting capability of the hydrostatic fluid feature is the ability to model cavities that are initially open when no hydrostatic pressure is applied. As the solution progresses, the cavity closes and the hydrostatic pressure is applied inside the closed cavity. The cavity opening is modeled with a contact pair and a dummy segment across the contact gap. The hydrostatic fluid pressure is applied using the birth time option when the cavity closes. An illustration of this capability is shown in Figure 4 below. As seen in the animation, the cavity on the right is initially open. When the two sides of the opening come into contact with each other, the cavity closes and hydrostatic pressure builds up within the cavity, resulting in the expansion of the cavity.

**Figure 4**

The ADINA Structures module provides state-of-the-art solution capabilities for linear and highly nonlinear analysis of solids and structures. Additionally, as shown in this Tech Brief, the ADINA Structures module provides solution capabilities for certain classes of problems involving fluids and solids or structures, such as:

- The analysis of fluid-filled cavities inside a solid or structure, where the fluid pressure in the cavity is assumed to be uniform at any point of time.
- The analysis of inviscid, irrotational fluids with no heat transfer. For example, see the Tech Brief on acoustic fluid elements and the analysis of LNG tanks.
- The modeling of hydrodynamic forces (e.g. the use of Morison's equation) in the analysis of offshore structures.

For more complex fluid-structure interaction problems, the ADINA FSI module can be used to solve the full Navier-Stokes equations for fluid flow.

*Keywords:*

Hydrostatic fluid loading, ideal gas law, uniform pressure, fluid cavities