Incident wave loading

One-dimensional incident wave loading is tested in this verification set. The model consists of a column of fluid 1 m long with a square cross-section of area equal to 10⁻⁴m². The length direction is the x-axis, while the cross-section is parallel to the y- and z-axes. In the axisymmetric case the column is oriented along the axial direction. The first-order element models consist of 100 elements for the quadrilateral cases and 200 elements for the triangular cases. The second-order element models consist of 50 and 100 elements for the quadrilateral and triangular cases, respectively. For all cases one element is used along the breadth and width directions.

A nonreflective boundary condition is imposed on one end of the column via impedance boundary conditions. The sound source is located at (−10, 0, 0) for the planar waves and at (−100000, 0, 0) for the spherical waves, while the standoff point is located at (0, 0, 0). The material properties of the fluid are the same as those of the surrounding medium. The material used is air with the following properties: density, ${\rho }_f=$ 1.21 kg/m³; bulk modulus, $K_f=$ 1.424 × 10⁵ Pa.

The sound source excitation is applied in two ways: through the pressure amplitude and through the corresponding acceleration amplitude. The pressure is applied as a ramp function beginning at zero and reaching a magnitude of 1.826 Pa at the end of 4.4 ms. The acceleration amplitude is applied through a step function with a magnitude of 1 m/s². Transient simulations are performed in both Abaqus/Standard and Abaqus/Explicit. The validity of the solution is checked by comparing the POR value at the first node with the expected value of 1.826 Pa at the end of the step.

The total wave formulation option is also tested. The acoustic solution under the specified incident wave loading obtained using the total wave formulation option is compared to the acoustic solution obtained while using the default scattered wave formulation option.

A similar model is also created to test the bubble loading, with water used as the material instead of air.

These are multiple-element tests that model sound sources of planar waves and spherical waves exciting traveling waves in ducts. Two cases are studied: a spherical wave source using an exponentially decaying time amplitude and a plane wave source using a sinusoidal amplitude. In both cases the total wave formulation is used and the standoff point of the incident wave loading is specified to be inside the finite element mesh. Consequently, at the start of the analysis the incident waves have already travelled into the finite element domain. These tests show that at the start of the first dynamic step in the analysis the acoustic field is properly initialized to the values of the incident wave field.

In the case of two-dimensional modeling single-element tests are used to verify incident wave loading on two-dimensional beam and pipe elements, where the wave source is located at (0.5, 10) for the planar waves and at (0.5, 100000) for the spherical waves. The single element for each case is placed along the x-axis with end points at (0, 0) and (1, 0). All nodes are completely fixed. The standoff point is at (0.5, 0). The beam element has a square cross-section of area 1 × 10⁻⁴ m², whereas the pipe has an outer diameter of 1.0 × 10⁻² m and the thickness of 1.0 × 10⁻³ m. The material properties for the beam are $E$ = 1.0 × 10⁶ Pa and $\rho$ = 1000 kg/m³. The properties of the surrounding medium are the same as those used in the previous section.

The loading is applied as a ramp function with a maximum value of 1000 Pa attained at the end of the step at 0.5 ms. The reaction forces at the element nodes are compared. The expected reaction force at each of the end nodes is 500 N for the linear elements. For quadratic elements the expected reaction force is 166.7 N at each of the end nodes and 666.7 N at the mid node.

The bubble loading is also tested but with water used as the material instead of air.

In the case of three-dimensional modeling for verification on three-dimensional beam and pipe elements in Abaqus/Explicit, a beam comprised of 50 beam (B31) or pipe (PIPE31) elements, placed along the x-axis with end points (−50, 0,0) and (50,0,0) is used. In both cases a cross-section of type pipe with an outer diameter of 2.5 m and a thickness of 0.1 m is used. The material properties for the beam are $E$ = 2.0 × 10¹¹ Pa and $\rho$ = 10000 kg/m³.

The source of the spherical wave, due to an under water explosion, is located at (0,−30,0); and the stand-off point is located at (0,−5,0). The wave load is applied over a cylindrical skin modeled with surface membrane elements (SFM3D4R) that is tied to the outer surface of the beam. Boundary conditions on the beam disallow any axial displacement and rotations along the y- and z-axis. The solution is computed for 0.1s.

These are single-element tests that model a sound source at (0.5, 0.5, 10) for the planar shells and at (0, −10) for the axisymmetric shells for the planar waves. For the spherical waves the source is moved to (0.5, 0.5, 100000) for the planar shells and to (0, −100000) for the axisymmetric shells. The planar shell is modeled to be in the X–Y plane with unit length on all sides. The standoff point is located at (0.5, 0.5, 0). In the axisymmetric case the shell is oriented along the radial direction and the standoff point is at (0, 0). The shell thickness is 10⁻⁴m. The shell material properties are the same as those of the beam in the previous section. The properties of the surrounding medium are kept the same as those used in the previous cases. All nodes are fixed completely. The loading is applied as a ramp function attaining a maximum of 1000 Pa at the end of the step at 0.5 ms. The reaction forces are compared with the expected values, which when summed should produce a total force of 1000 N.

A similar model is also created to test the bubble loading, with water used as the material instead of air.

These tests use exactly the same geometry as that used in the acoustic element tests, except that the length is reduced to 0.1 m. Consequently, 10 and 20 first-order elements are used in the quadrilateral and triangular cases, respectively; and 5 and 10 second-order elements are used for the quadrilateral and triangular cases, respectively. The sound source is at (−10, 0) for the planar waves and at (−100000, 0, 0) for the spherical waves. All nodes are fixed in the y-direction, while the end nodes on the surface further away from the source are fixed additionally in the x-direction. The stresses in the elements are compared with those obtained using the equivalent distributed surface load.

A similar model is also created to test the bubble loading, with water used as the material instead of air.

One-dimensional incident wave loading is tested for coupled analysis in this verification set. When solid and beam elements are coupled with the acoustic elements, the sound source is located at (−10, 0, 0) for the planar waves and at (−100000, 0, 0) for the spherical waves. For coupling with shell elements the sound source is located at (0, 0, 10) for the planar waves and at (0, 0, 100000) for the spherical waves. For all the axisymmetric cases the sound source is located at (0, −10) for the planar waves and at (0, −100000) for the spherical waves. The standoff point is located at (0, 0, 0).

One acoustic element is used for the coupling analysis. The two-dimensional acoustic element has a length and width of 1 m and a thickness of 10^–4 m. The three-dimensional acoustic element has unit length on all sides. The material properties for the acoustic elements are as follows: density, ${\rho }_f=$ 1.21 kg/m³; bulk modulus, $K_f=$ 1.424 × 10⁵ Pa. The material properties of the surrounding medium are the same as those of the fluid. The planar shells are modeled in the X–Y plane with a surface lying on one face of the acoustic element. The shell element thickness is 10^–4 m. The beam elements are modeled parallel to the y-direction and lying on one edge of the two-dimensional acoustic element. The beam has a square cross-section area of 10⁻⁴ m². Solid elements are modeled with the length direction as the x-axis and the other two directions parallel to the y- and z-axes; they are placed adjacent to the acoustic elements. In axisymmetric cases the elements are oriented in the axial direction. The material properties of the solid and structural elements are the same as those used in the previous cases.

All nodes are kept fixed for the beam and shell elements. For the solid elements all nodes are fixed in the y-direction, and the nodes that are further away from the tied surface are fixed additionally in the x-direction. For the acoustic elements the loading is applied as a ramp function attaining a maximum of 2.0755 Pa at the end of the step at 5 ms. Additionally, pressure is applied for the solid and structural elements as a ramp function with a maximum of 5 Pa at the end of the step. The results are compared with the expected values of reaction forces and POR.

Two similar models are also created to test the bubble loading, with water used as the material instead of air.

These are single-element tests that model a sound source at (0.0, 0.0, 10.0) for the spherical waves and a reflecting surface 5 m directly above the sound source. The standoff point is located at (0.0, 0.0, 0.0). The planar shell is modeled in the X–Y plane with unit length on all sides. The shell thickness is 10^–4 m. All nodes are fixed for the planar shells. The shell material properties are as follows: E=10⁶ Pa and $\rho$=1000 kg/m³. The three-dimensional acoustic element is modeled with one face of the element on the X–Y plane and has unit length on all sides. The material properties are the same as those used in the previous case. The surrounding medium has the following material properties: density, ${\rho }_f$=100 kg/m³; bulk modulus, $K_f$=10⁸ Pa. The loading is a step function with pressure magnitude of 1000 Pa for planar shells and 415.09517 Pa for acoustic elements. Four different properties of the reflecting surface are considered for the tests. For planar shells the reaction forces are compared with the expected values. For acoustic elements POR values are compared.

These are single-element tests that model a sound source at (0.0, 10.0, 10.0) for the direct-path waves and a reflecting surface 20 m directly below the sound source. The standoff point is located at (0.0, 0.0, 0.0). The loading amplitude is a step function with pressure magnitude of 1000 Pa for the planar shells and 1.0 Pa for the acoustic elements.

The planar shell is modeled in the X–Y plane with unit length on all sides. The shell thickness is 10^–4 m. All nodes are fixed for the planar shells. The shell material properties are as follows: E=10⁶ Pa and $\rho$=1000 kg/m³.

The three-dimensional acoustic element is modeled with one face of the element on the X–Y plane and has unit length on all sides. The acoustic medium has the following material properties: density, ${\rho }_f$=1.0 kg/m³; bulk modulus, $K_f$=1.6 × 10 ⁵ Pa, resulting in a speed of sound of 400 m/s.

For planar shells the reaction forces are compared with the expected values. For acoustic elements POR values are compared.

These are simple tests to verify the application of planar, spherical, and diffuse incident wave fields in steady-state dynamics.