Smoothed particle hydrodynamic analysis

This model analyzes the impact interaction between a flying object and a rotating airplane engine blade. The airplane engine blade is modeled using 960 S4RS shell elements. A set of nodes closer to the turbine hub are kinematically coupled to a reference node situated at the center of the hub. A constant angular velocity of $5.466\times {10}^2$ rad/s is applied at the reference node about the z-axis. The engine blade is modeled with an elastic-plastic material with Young's modulus $E=210$ GPa, Poisson's ratio $\nu =0.3$, density $\rho =4.0\times {10}^3$ kg/m³, and isotropic hardening. The flying bird is modeled using 4160 PC3D elements. The bird material is modeled using a tabular equation of state (EOS) material with a tensile failure strength of 94 MPa and a density of $\rho =1.0\times {10}^3$ kg/m³. The radius of the cross-section of the cylinder modeling the bird is 0.04 m, and the height of the cylinder is 0.076 m. The contact interaction between the surfaces of the bird object and the shell structure is defined through contact inclusions.

The initial configuration of the model is shown in Figure 1.

An intermediate deformed configuration of the airplane engine blade and the bird system is shown in Figure 2.

Overall the model, the material properties, and the loading conditions are the same as in Bird strike on an airplane engine blade. The only exception is that the bird is first modeled with C3D8R elements rather than with PC3D elements. A strain-based criterion is used to convert each continuum element to eight SPH particles. The contact interaction between the internally generated particles and the shell structure is defined automatically from the user-defined contact inclusions.

The initial configuration of the model is shown in Figure 3.

An intermediate deformed configuration of the airplane engine blade and the bird system is shown in Figure 4.

This model analyzes the impact and mixing of two liquids with the same material properties. The spherical liquid drop and the liquid in the container are modeled using 3544 and 9000 PC3D elements, respectively. Both liquids are defined using an EOS material of type USUP modeling a linear equation of state. The parameters used in this material model are $c_0=0.111803398875$ mm/s, $s=0.0$, and ${\mathrm{\Gamma }}_0=0.0$. To increase the stable time increment, the density of the liquid is artificially defined as $\rho =$1 tonne/mm³. The height of the container is 5 mm. The horizontal cross-section of the container has a square shape with a side length of 15 mm. The lateral and bottom walls of the container are modeled as S4R shell elements. The contact interaction between the liquid and the shell structure is defined through contact inclusions.

The initial configuration and an intermediate configuration are shown in Figure 5 and Figure 6.

This model analyzes the impact interaction between a liquid modeled using the SPH technique and a rigid solid structure. The block of liquid is modeled using 53040 PC3D elements. The material model of the liquid used is an EOS material of type USUP modeling a linear equation of state. The material parameters used are $c_0=1.5\times {10}^6$ mm/s, $s=0.0$, and ${\mathrm{\Gamma }}_0=0.0$. A failure strength of 2 MPa is defined for this EOS type material. To model the figurehead, 4084 R3D3 rigid elements are used. The initial velocity of the liquid is 3000 mm/s along the y-direction toward the figurehead. The confining box has a dimension of 800 mm × 800 mm × 500 mm, and it is modeled using 48 R3D4 rigid elements. The contact interaction between the liquid and the surfaces of the figurehead and the confining box is defined through contact inclusions with the no-friction surface interaction.

The initial and intermediate configurations are shown in Figure 7 and Figure 8.

This problem analyzes the temperature-related failure of a figurehead statue modeled using the SPH technique. The figurehead statue is modeled using 8252 PC3D elements, and it is characterized by a temperature-dependent elastic-plastic material mode via field variable dependencies. The Young's modulus, $E$, is equal to 2 MPa when the non-dimensional field variable is equal to 1.0, and it is equal to 0.8 MPa when this variable changes to 2.0. The dependence of the plastic properties on the temperature is given via tabular data. The density of the material is defined as $\rho =1.0\times {10}^{-8}$ tonne/mm³. Fifty R3D4 elements are used to model the bottom and the lateral walls. The melting process of the figurehead statue is accelerated after a sudden rise of the temperature during the dynamic analysis. The contact interaction between the solid statue and the rigid wall is defined through contact inclusions.

The initial and intermediate configurations are shown in Figure 9 and Figure 10.

This model analyzes the impact interaction between a figurehead and solid walls. The figurehead is modeled using 8252 PC3D elements. The material model used for this figurehead is an EOS material of type USUP modeling a linear equation of state. The material parameters used are $c_0=1.405\times {10}^5$, $s=0.0$, and ${\mathrm{\Gamma }}_0=0.0$. A linear viscous shear behavior is defined for this hydrodynamic material through tabular data. A tensile failure strength of 10 MPa is also defined for this material. The density of the figurehead is set to $\rho =9\times {10}^{-10}$ tonne/mm³. Fifty R3D4 elements are used to model the bottom and the lateral walls. The initial velocity of the figurehead is set to 1.0$\times {10}^3$ mm/s toward the lateral wall. The figurehead then follows a parabolic path under gravitational forces until it strikes the wall. After the impact, the figurehead is smashed onto the lateral wall and then crashes into the corner edge because of complete material failure. The contact interaction between the figurehead and the rigid wall is defined through contact inclusions using rough friction to describe the frictional interactions.

The initial configuration and an intermediate configuration of the figurehead and the rigid walls are shown in Figure 11 and Figure 12.

This model analyzes the impact interaction between a high-velocity projectile and a solid plate. The solid plate has a dimension of 400 mm × 400 mm × 12 mm. A circular part with a radius of 100 mm in the center of the plate is modeled using 102726 PC3D elements, and the remaining part of the plate is modeled using 9312 C3D8R elements. The length and radius of the cylindrical rigid solid projectile are 25 mm and 8.4 mm, respectively. The initial speed of the projectile is set to 1000 m/s. The material used for the plate is a steel with Young's modulus $E=2.1\times {10}^5$ MPa, Poisson's ratio $\nu =$ 0.3, and density $\rho =7.8\times {10}^{-9}$ tonne/mm³. The plate is modeled as an elastic-plastic material with rate-dependent hardening. Ductile and shear damage are evolved based on an energy criterion. The interaction between the rigid projectile and the solid plate is defined using frictional contact with a friction coefficient of 0.3.

The initial configuration of the model is shown in Figure 13, and an intermediate deformed configuration cross-section is shown in Figure 14.