This example shows a drop test of a fluid-filled plastic bottle.
The following
Abaqus
features are demonstrated:
using the Eulerian-Lagrangian contact formulation to simulate a highly
dynamic event involving a fluid material (modeled using Eulerian elements)
interacting with structural boundaries (modeled using Lagrangian elements),
using the smoothed particle hydrodynamic
(SPH) method to analyze a highly dynamic event
in a purely Lagrangian environment, and
converting continuum finite elements to
SPH particles to analyze the highly dynamic
event.
Simulation is commonly used in the consumer packaging industry to reduce the
time and cost associated with physical prototyping. Drop tests, which simulate
an object falling and impacting a hard surface, are often used to investigate
the object's response under harsh handling conditions.
This example involves a fluid-filled plastic bottle falling from a height of
roughly 300 mm onto a flat, rigid floor. The bottle, as shown in
Figure 1,
is a rectangular jug made of high-density polyethylene. The bottle is filled
almost completely (about 95%) with water. A realistic simulation for the bottle
must account for both the exterior forces on the bottle from the floor impact,
as well as the interior forces of the water pushing against the bottle.
Resulting stresses and strains in the bottle can be used to determine its
structural feasibility.
Geometry
Figure 1
shows the pertinent dimensions for the bottle and cap model. The bottle has a
uniform thickness of 0.5 mm, with the exception of the rim around the bottle's
mouth; the rim is 0.65 mm thick. The bottle's cap is modeled as a separate part
instance and positioned on the mouth of the bottle; the cap is uniformly 1 mm
thick.
The bottle strikes the floor at a skew angle, with one of the bottom corners
experiencing the initial impact.
Figure 2
shows the assembled model. Initially the water in the bottle is distributed
according to a gravitational response; that is, the boundary of the water is
parallel to the horizontal floor, not to the bottom of the skewed bottle.
Materials
The bottle is constructed of high-density polyethylene that follows an
isotropic plastic hardening model.
The water is treated as a nearly incompressible, nearly inviscid Newtonian
fluid.
Boundary conditions and loading
The entire model is subject to a gravity load. The rigid floor is fixed in
place.
Interactions
The bottle contacts three different model components over the course of the
analysis: the floor, the bottle cap, and the water within the bottle. All of
these contact interactions are assumed to be frictionless.
Abaqus modeling approaches and simulation techniques
The primary challenge to solving this problem is the highly transient
fluid-structure interaction between the water and the bottle. In this example
two methods are studied. The coupled Eulerian-Lagrangian
(CEL) analysis technique in
Abaqus/Explicit
is well suited to handling problems of this nature. The
SPH method can be used to model the violent
sloshing associated with the impact.
Summary of analysis cases
Case 1
Coupled Eulerian-Lagrangian analysis.
Case 2
SPH
analysis.
Case 3
Finite element conversion to
SPH particle analysis.
Mesh design
The bottle is imported as an orphan mesh of S3R and S4R elements. The cap geometry is meshed with S4R elements.
Figure 3
shows the assembled mesh for the bottle and cap. The floor is meshed with
surface elements of type SFM3D4R; a rigid body constraint is subsequently applied to these
elements (see
Constraints
below).
Materials
An elastic-plastic material definition is used for the polyethylene in the
bottle, with the isotropic hardening curve defined by the data points in
Table 1;
the onset of plastic yielding occurs at 8.618 N/mm2, and failure
occurs at a strain of 0.59. Damage is incorporated using a ductile damage
definition. The high-density polyethylene has a density of 8.76 ×
10−7 kg/mm3, Young's modulus of 903.114 N/mm2,
and Poisson's ratio of 0.39.
The water is modeled using the linear
Hugoniot form of the Mie-Grüneisen equation of state; the equation parameters
appear in
Table 2.
The water has a density of 9.96 × 10−7 kg/mm3 and bulk
modulus of 2.094 GPa.
Boundary conditions
An encastre boundary condition is applied to the reference point of the
rigid floor to fix it in place for the duration of the analysis.
Loads
A gravity load is applied to the bottle, cap, and water instances. An
acceleration of −9800 mm/s2 is applied in the
z-direction.
Predefined fields
Instead of simulating the full dropping event from the initial position, the
bottle, cap, and water instances are positioned close to the floor and
prescribed an initial velocity predefined field. An initial velocity of −2444
mm/s in the z-direction corresponds to the speed that
would be attained by an object falling about 300 mm from rest under typical
gravitational acceleration.
Constraints
A rigid body constraint applied to the floor part instance makes the floor a
simple, undeformable surface.
Interactions
General contact is defined for the model. General contact enforces
interactions between the bottle and other Lagrangian components, such as the
cap and floor. The default frictionless hard contact property governs all
interactions.
In an actual bottle, the cap would be firmly attached to the neck of the
bottle. This interaction between the bottle and the cap is considered
insignificant for the purposes of this analysis. Instead of adding undue cost
to the analysis by modeling a threaded connection or tie constraint between the
two parts, the cap is allowed to freely detach from the bottle during the
simulation.
Output requests
Displacements, velocities, and accelerations are requested for the bottle
and cap. Logarithmic strain (LE) and equivalent plastic strain (PEEQ) are requested for the bottle to assess its structural
response. Damage initiation criteria (DMICRT) and element status (STATUS) are also requested for the polyethylene bottle to track
potential failure in these components. Finally, contact stresses (CSTRESS) and contact forces (CFORCE) are requested for the Lagrangian part instances.
Case 1 Coupled Eulerian-Lagrangian analysis
This case shows the use of the Eulerian-Lagrangian contact formulation to
simulate the bottle drop with the fluid material (modeled using Eulerian
elements) interacting with structural boundaries (modeled using Lagrangian
elements).
Analysis types
The full simulation is conducted in a single explicit dynamic step lasting
0.05 s.
A few distorted Lagrangian elements in the model are controlling the stable
time increment, thus dictating the time to complete the analysis. Using the
semi-automatic mass scaling method, the time increment can be increased to
7.528 × 10−7 s, reducing the analysis cost by nearly half, while
increasing the total mass by only 0.11 percent. Alternatively, using the mass
adjustment method, the mass can be redistributed among the elements of the
bottle to achieve the same time increment without affecting the total mass of
the bottle. A large number of elements have mass in excess of what is required
to achieve the above stable time increment. Only 0.12 percent of this excess
mass is redistributed to the remaining elements to raise their time increment
to the specified value. In both methods, the change in the mass distribution is
very small and does not significantly influence the analysis results.
Analysis techniques
The Eulerian element formulation allows the analysis of bodies undergoing
severe deformation without the difficulties traditionally associated with mesh
distortion. In an Eulerian mesh material flows through fixed elements, so a
well-defined mesh at the start of an analysis remains well-defined throughout
the analysis. Although Eulerian material boundaries are more approximate than
traditional Lagrangian element boundaries, the Eulerian formulation allows you
to capture extreme deformation phenomena such as fluid flow. The water is
modeled using an Eulerian element domain. The bottle—which, although
significantly stiffer than the water, is not rigid—is modeled using traditional
Lagrangian shell elements. The general contact algorithm in
Abaqus/Explicit
tracks and enforces contact between the Eulerian material boundary and the
Lagrangian elements, enabling effective simulation of the fluid-structure
interaction.
Mesh design
The Eulerian mesh, which serves as the domain through which the water
material can flow, is a 300 × 250 × 200 mm rectangular prism of EC3D8R elements. Each Eulerian element is a regular cube measuring 5 mm
on an edge. The Eulerian mesh completely encompasses the bottle and cap, and it
extends slightly below the floor. Any interface that is expected to experience
Eulerian-Lagrangian contact must be located within the Eulerian mesh; once an
Eulerian material passes beyond the boundaries of the Eulerian mesh, it is lost
to the simulation and contact is not enforced. The overlapping of Lagrangian
and Eulerian elements is acceptable because these two element types do not
interact with each other. Lagrangian elements interact only with Eulerian
material within the mesh. The initial position of this Eulerian material must
be defined within the Eulerian mesh, as discussed in
Initial conditions
below.
Initial conditions
Because an Eulerian mesh is void of any material by default, the desired
distribution of material within the Eulerian mesh must be specified using an
initial condition. This distribution is defined using the concept of an
Eulerian volume fraction, or the percentage of an element that is occupied by a
given material. For each Eulerian element that initially contains material, an
initial Eulerian volume fraction for that material must be specified.
Abaqus
evaluates all of the element volume fractions to determine the distribution and
boundaries of Eulerian materials within the Eulerian mesh.
Interactions
The contact between the bottle and the Eulerian material initially
positioned inside of the bottle is also modeled using the general contact
algorithm.
Output requests
In addition to the field and history output requests listed above, the
Eulerian volume fraction output variable (EVF) is requested as field output to visualize geometric results.
Case 2 SPH analysis
This case shows the use of a smoothed particle hydrodynamic
(SPH) analysis to analyze the fluid-filled
bottle drop in a purely Lagrangian environment.
Analysis types
The full simulation is conducted in a single explicit dynamic step lasting
0.05 s.
Analysis techniques
The water is modeled using continuum pseudo-particles. Internally,
Abaqus/Explicit
determines automatically every increment of the analysis which are the active
neighbors associated with a given particle of interest in order to apply the
SPH formulation. Since the internally
determined connectivity is allowed to change every increment, the method
robustly handles the severe deformations associated with the sloshing water.
The other Lagrangian bodies (bottle, cap, and floor) are modeled the same as in
the coupled Eulerian-Lagrangian method.
Mesh design
The pseudo-particles are modeled using PC3D elements that are spaced in a relatively uniform fashion every 5
mm in all directions. These one-node elements fill only the space initially
occupied by the water just before the impact. Therefore, fewer PC3D elements are needed when compared to the number of EC3D8R elements in the coupled Eulerian-Lagrangian method.
Interactions
A node-based surface associated with the water pseudo-particle is included
in the contact domain to model the interactions between the water and the
bottle.
Case 3 Finite element conversion to SPH
particles analysis
This case shows the use of the technique for converting reduced-integration
continuum elements to SPH particles.
Analysis types
The full simulation is conducted in a single explicit dynamic step lasting
0.05 s.
Analysis techniques
The water is modeled using reduced-integration continuum C3D4 elements. A time-based criterion is used to trigger the
conversion from a user-defined mesh to SPH
particles (PC3D elements) at the beginning of the analysis. Upon conversion the
continuum elements modeling the fluid will become inactive (deleted from the
mesh) while the PC3D elements get activated. Since all particles get activated at the
same time, the SPH functionality will model
the fluid as described in Case 2.
Mesh design
The water is modeled using reduced-integration continuum C3D4 elements using regular preprocessing techniques; the mesh is
shown in
Figure 4.
Thus, cumbersome meshing approaches based on unconnected particles are avoided.
Interactions
Contact between the fluid and the bottle is defined by including an
element-based surface that includes the fluid's interior and its initial
external surface in the general contact domain. Internally,
Abaqus/Explicit
generates a node-based surface for the generated
SPH particles that will be also included in
the contact domain.
Discussion of results and comparison of cases
The results of the drop test in the CEL
analysis appear in
Figure 5.
The water contributes significantly to the behavior of the bottle: the
bottle flexes and bulges as the water sloshes.
Figure 6
shows the logarithmic strains in the bottle. Strains appear in the sides of the
bottle as they bulge outward, but these strains are largely recovered when the
water sloshes upward. The most significant strains occur on the bottom of the
bottle, along the bottle-floor interface. Despite the deformation caused by the
floor and the water, the damage criteria for the polyethylene is not met at any
location on the bottle.
A comparison between the CEL method and the
SPH method at different stages during impact
is illustrated in
Figure 7
through
Figure 10.
The comparison between the two methods is quite reasonable.
The effect of sloshing water can also be seen in the reaction force on the
floor surface. The reaction force in the z-direction is
plotted in
Figure 11
using a Butterworth filter with a cutoff frequency of 2000. The reaction force
peaks sharply at approximately 0.02 s, which corresponds roughly with the
appearance of strains in the sides of the bottle. After this point, the water
surges upward, offsetting the downward momentum of the bottle and reducing the
reaction forces in the floor. Good agreement between the two methods used is
observed.
The significant deformation of the fluid in this problem clearly influences
the analysis results. The coupled Eulerian-Lagrangian and
SPH techniques in
Abaqus/Explicit
provide an effective way to realistically capture the complexity of the
fluid-structure dynamics.
Files
The basic system of units in this problem is kilograms (kg), millimeters
(mm), and seconds (s). Under this system, the base unit of force is kg
mm/s2, or 10−3 newtons; values for all force-derived
quantities are entered in the sample files using this base unit. Values for all
force-derived quantities are reported in the documentation in units of standard
newtons (kg m/s2) and scaled appropriately.
Finite element conversion to SPH particles
input file for the model.
Tables
Table 1. Isotropic plastic hardening data for polyethylene material.
Yield stress (N/mm2)
Plastic strain
8.618
0.0
13.064
0.007
16.787
0.025
18.476
0.044
20.337
0.081
24.543
0.28
26.887
0.59
Table 2. Material parameters for water.
Parameter
Value
Density ()
9.96 × 10−7 kg/mm3
Viscosity ()
1 × 10−8 N s/mm2
1.45 × 106 mm/s
s
0
0
Figures
Figure 1. Bottle used in the drop test. All dimensions are in
millimeters. Figure 2. Assembly of the bottle drop model; the water is instanced inside of
the bottle. Figure 3. Mesh of the bottle and cap. Figure 4. Tetrahedral mesh to model the fluid in the bottle in the undeformed
configuration. Figure 5. Deformed shape plots of the CEL model
at 0.01 s intervals. Figure 6. Logarithmic strains in the bottle during deformation in the
CEL analysis. Figure 7. Comparison between coupled Eulerian-Lagrangian and
SPH results shortly after the impact. Figure 8. Comparison between coupled Eulerian-Lagrangian and
SPH results after full contact with the
floor. Figure 9. Comparison between coupled Eulerian-Lagrangian and
SPH results in the middle stages of the
analysis. Figure 10. Comparison between coupled Eulerian-Lagrangian and
SPH results towards the end of the
analysis. Figure 11. Reaction force (filtered) in the floor.
