###Acoustic Problem###

Keywords: acoustic propogation, Helmholtz Operator, High-Order Approximation, Exterior Boundary value problem, Non-reflection Boundary Condition

Problem description

Three acoustic problems implemented with analytical solutions, as well as scalar L2 and H1 norm error estimation techniques which can perform convergence tests or for the usage of h&p adaptivity. Due to the large amounts of mathematical formulations, for details of acoustic problems and derivation of analytical solutions please see [1].

Theoretical background

For the details of SommerFeld boundary condition and Baliss-Turkel Boundary condition see [3], while using hierarchic finite element [2].

Input files

The CUBIT journal file required to generate the input model can be downloaded from here (same file is used for all type of boundary conditions including linear displacement, traction and periodic):

• plane_wave_cube.jou (CUBIT journal)

• impinging_cyclinder.jou (CUBIT journal)

• impinging_sphere.jou (CUBIT journal)

Material properties

Scalar

• Angular frequency

• Speed of Sound

• Amplitude of Incident Wave

Finite Element Model

The finite element model consists of a 4-node tetrahedral elements as shown in Figure below.

Figure. 4-node tetrahedral

Wave Guide Problem

Execution commands

First run

make -j2 && mpirun -np 5 ./best_approximation -my_file plane_wave_cube.cub -my_is_partitioned false -wave_number 2 -wave_direction 1,0,0 -analytical_solution_type plane_wave -save_postproc_mesh true -ksp_type fgmres -pc_type lu -pc_factor_mat_solver_package superlu_dist -ksp_monitor -my_order 5 -my_max_post_proc_ref_level 0 -add_incident_wave false wave_guide_angle 0

• -wave_number scalar (angular frequency/speed of sound)
• -wave_direction 3*3 unit vector (direction of the incident wave)

• -analytical_solution_type characters chosen from:

  "hard_sphere_incident_wave"

  "soft_sphere_incident_wave"

  "plane_wave"

  "hard_cylinder_scatter_wave"

  "soft_cylinder_scatter_wave"

  "incident_wave"

  "no_analytical_solution" (type of the analytical solution)

• -add_incident_wave bool (whether you require a scatterer potential or total potential)

• -power_of_incident_wave scalar (the amplitude of incident wave)

then you can run

mpirun -np 5 ./fe_approximation -my_file analytical_solution.h5m -my_is_partitioned false -wave_number 2 -wave_direction 1,0,0 -analytical_solution_type plane_wave -save_postproc_mesh true -ksp_type fgmres -pc_type lu -pc_factor_mat_solver_package mumps -ksp_monitor -my_order 2 -my_max_post_proc_ref_level 0 -add_incident_wave false wave_guide_angle 0

mpirun -np 2 ./error_norm -my_file fe_solution.h5m -norm_type l2 -relative_error false -ksp_type fgmres -pc_type lu -pc_factor_mat_solver_package superlu_dist  -ksp_monitor  -my_order 1 -my_max_post_proc_ref_level 0

Figure 0. Example 1 2D Wave Guide Problem

Figure 1. Example 1 Real part of total potential for wave guide problem K=10

Figure 2. Example 1 Imag part of total potential for wave guide problem K=10

Figure 3. Example 1 log-log h-convergence of l2 relative error versus DOFs for wave guide problem K=10

Figure 4. Example 1 log-log p-convergence of l2 relative error versus DOFs for wave guide problem K=10

Cylinder Scatterer Problem

Figure 5. Example 2 Absolute value of total potential for impinging Cylinder problem K=10

Sphere Scatterer Problem

Sub-marine Scatterer Problem

References

1. Ihlenburg, F. Finite element analysis of acoustic scattering Springer Science & Business Media. 1998

2. M. Ainsworth and J. Coyle. Hierarchic finite element bases on unstructured tetrahedral meshes. International Journal for Numerical Methods in Engineering, 58(14):2103– 2130, 2003.

3. Strouboulis T, Babuska I, Hidajat R.The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment. Computer Methods in Applied Mechanics and Engineering 197.5 (2008): 364-380.

• Created by Thomas Xuan Meng
• any difficulties or suggestions email cmatgu@googlegroups.com