The submerged infinite cylinder problem

The infinite cylinder excited by a plane wave is a two-dimensional problem with a single plane of symmetry. The half-model for this problem is shown in Figure 1. Plane strain boundary conditions are imposed along the axis of the cylinder, and symmetry boundary conditions are imposed on the x–z plane. Boundary conditions are imposed on the structural model. A consistent set of boundary conditions need not be given for the fluid since they are applied by default. The structure is enclosed in a fluid mesh that models the surrounding fluid. Two different meshes are chosen, and the results are compared. The infinite cylinder is assumed to be fully submerged in the fluid so that free surface effects are negligible. The cylindrical shell is assumed to be made of steel, and the fluid is assumed to be water; however, the material properties have been nondimensionalized. The coupling between the structure and the fluid is enforced using a tie constraint, which does not require compatible meshes. The mesh density for the fluid is chosen such that the shock is captured accurately. The condition for non-reflection is applied on the outer surface of the fluid mesh via the predefined circular radiation boundary condition.

An explosion occurs on the x-axis far from the structure. The incident wave loading is applied at the interface of the structure and the fluid as part of the history data, specifying the location of the charge and the standoff point. Since the front is planar, the charge location is used only to compute the direction of the incoming shock. The standoff point is chosen as the point on the structure closest to the charge, so that the simulation begins when the front is just about to impinge on the structure. The pressure profile of the shock wave measured at the charge standoff point is given by an amplitude curve. For this problem the pressure profile is a step function with a magnitude of 1.0 × 10⁻⁴.

The half-cylinder is modeled with 4-node quadrilateral shell elements. In both models the structure is meshed with 36 S4R elements. The fluid meshes are different in the models displaying variations of mesh density. Fluid meshes are modeled with AC3D8R elements. The analyses are run with double precision in Abaqus/Explicit, using direct user control for the time interval. The bulk viscosities and the interval sizes have been specified to optimize the efficiency of the solution, by reducing the run time and capturing the shock accurately.

The infinite cylinder is excited by a shock wave from a charge that is relatively close to the structure. The effect of the pressure wave reflecting off the bottom surface is included. Bottom surface effects are treated in Abaqus using imaging techniques where the incident pressure wave is made up of both primary and image contributions with the appropriate time delays automatically calculated. The analysis is performed in Abaqus/Explicit using both the total and scattered wave formulations.

The cylinder geometry and fluid properties are the same as those defined in the first part of this example. While the structure is symmetric, the loading is not. Therefore, symmetry is not used in the analysis. A cross-section of the full cylinder model is shown in Figure 4. A single charge is located on the x-axis 10 length units away from the cylinder (z) axis. The bottom surface is located 4.6904 length units below the cylinder axis. It is assumed that 80% of the incoming wave is reflected off the “soft” bottom surface. The charge location is included as data with the incident wave property. The bottom surface location is specified using an incident wave reflection. The reflective properties of the surface are converted into equivalent impedance properties and specified using an impedance property. The model consists of 72 4-node quadrilateral shell elements and 2880 AC3D8R fluid elements. The standoff point is placed along the x-axis, at the point where the structure and fluid come into contact.

A two-charge model, wherein the bottom surface is represented by a second “image” charge in addition to the original primary charge, is also considered. The image charge is located at the same x-direction coordinate used for the primary charge but below it by twice the distance to the bottom surface. The bottom surface definition is intentionally excluded from this model. Multiple charges are represented by multiple incident wave loading definitions within an analysis step. To specify the time delay due to reflection correctly, the distance between the image charge and its standoff point is set equal to the distance between the primary charge and its standoff point. In addition, the standoff point is located on the line joining the image charge to the center of the circle. To simulate the partial reflection requirement, the image charge magnitude is scaled by the reflection coefficient of 0.8.