Problem with analytical solution
Elements tested
Problem description
A column of poroelastic material fixed in the lateral directions and
attached to a wall is loaded at the tip by a harmonic pressure in a direct
steady-state dynamic analysis. The analytical solution for the two
compressional waves is available in
Allard and Atalla (2009), sec. 6.5–6.6.
Finite element solutions are obtained for three models having 10, 100, and 1000
finite elements in this one-dimensional problem to verify convergence to the
analytical solution. To verify tie constraints to connect
poroelastic-to-poroelastic elements, each model is subdivided into two parts
with the same material.
Material:
The skeleton is linear poroelastic with:
| Porosity, 0.94
|
| Tortuosity, 1.06
|
| Young's modulus, 4.4 × 106
|
| Poisson's ratio, 0
|
| Density, 130
|
| Structural material bulk modulus, 1.0 ×
1010
|
| Pore fluid, air
|
| Density, 1.177
|
| Static flow resistivity, 40000
|
| Viscous characteristic length, 56 × 10-6
|
| Thermal characteristic length, 110 × 10-6
|
| Fluid dynamic shear viscosity, 1.846 ×
10-5
|
| Ambient fluid standard pressure, 1.0 ×
105
|
| Ambient fluid heat capacity ratio, 1.4
|
| Prandtl number, 0.707
|
The MKS unit system is used. These parameters are used directly with
the Biot-Johnson acoustic medium porous model. For the Biot-Atalla model the
equivalent fluid complex density and fluid complex bulk modulus are provided
instead.
Boundary conditions:
The mesh is aligned along the Z-direction.
Lateral X- and
Y-displacements are applied at all nodes, and all
three displacements at the wall nodes are fixed. External pressure is specified
as a boundary condition for the pore pressure and as an external pressure load
at the tip element face.
Results and discussion
A solution is obtained for several excitation frequencies in accordance with
Allard and Atalla (2009). With mesh
refinement, asymptotic pointwise convergence to the analytical solution is
for the tip displacements and pore pressure at the wall in both real and
imaginary components. Results for the C3D8A elements for two excitation frequencies, 300 and 1300 Hz, are
presented below.
| Real tip
displacements, 300 Hz, analytical solution = -2.878E-08
|
| Number of elements
|
Error
|
| 10
|
-5.94E-11
|
| 100
|
-5.92E-13
|
| 1000
|
-5.E-15
|
| Imaginary tip
displacements, 300 Hz, analytical solution = -8.784E-09
|
| Number of elements
|
Error
|
| 10
|
-6.07E-11
|
| 100
|
-6.09E-13
|
| 1000
|
-6.5E-15
|
| Real pore pressure
at the wall, 300 Hz, analytical solution = -7.765E-02
|
| Number of elements
|
Error
|
| 10
|
-2.14E-03
|
| 100
|
-2.14E-05
|
| 1000
|
-2.11E-07
|
| Imaginary pore
pressure at the wall, 300 Hz, analytical solution = -2.768E-01
|
| Number of elements
|
Error
|
| 10
|
-9.44E-04
|
| 100
|
-9.59E-06
|
| 1000
|
-1.1E-07
|
| Real tip
displacements, 1300 Hz, analytical solution = -1.077E-08
|
| Number of elements
|
Error
|
| 10
|
1.77E-09
|
| 100
|
1.51E-11
|
| 1000
|
1.53E-13
|
| Imaginary tip
displacements, 1300 Hz, analytical solution = -6.522E-09
|
| Number of elements
|
Error
|
| 10
|
3.79E-10
|
| 100
|
3.19E-12
|
| 1000
|
3.25E-14
|
| Real pore pressure
at the wall, 1300 Hz, analytical solution = -6.516E-02
|
| Number of elements
|
Error
|
| 10
|
5.13E-03
|
| 100
|
4.32E-05
|
| 1000
|
4.36E-07
|
| Imaginary pore
pressure at the wall, 1300 Hz, analytical solution = 6.955E-03
|
| Number of elements
|
Error
|
| 10
|
8.41E-03
|
| 100
|
7.84E-05
|
| 1000
|
7.92E-07
|
- aa2009anal.inp
- C3D8A elements, 10 element mesh.
- aa2009anal2.inp
- C3D8A elements, 100 element mesh.
- aa2009anal3.inp
- C3D8A elements, 1000 element mesh.
- aa2009analc3d6a.inp
- C3D6A elements, 20 element mesh (each cube of the 10 element mesh is
subdivided into two wedges).
- aa2009analc3d6a2.inp
- C3D6A elements, 200 element mesh (each cube of the 100 element mesh is
subdivided into two wedges).
- aa2009analc3d6a3.inp
- C3D6A elements, 2000 element mesh (each cube of the 1000 element mesh
is subdivided into two wedges).
- aa2009analc3d4a.inp
- C3D4A elements, 60 element mesh (each cube of the 10 element mesh is
subdivided into six tetrahedra).
- aa2009analc3d4a2.inp
- C3D4A elements, 600 element mesh (each cube of the 100 element mesh is
subdivided into six tetrahedra).
- aa2009analc3d4a3.inp
- C3D4A elements, 6000 element mesh (each cube of the 1000 element mesh
is subdivided into six tetrahedra).
- aa2009analconstr2.inp
- C3D8A elements, 10 element mesh (5 plus 5 matched) with tie
constraints.
- aa2009analconstr4.inp
- C3D8A elements, 15 element mesh (10 plus 5 unmatched) with tie
constraints.
References
- Allard, A. F., and N. Atalla, Propagation
of Sound in Porous Media: Modelling Sound Absorbing
Materials, John
Wiley & Sons,
Ltd., Second
Edition, 2009.
Testing compatibility boundary conditions
Problem description
The problem is similar to the problem above but does not need an analytical
solution. It tests poroelastic-to-elastic and poroelastic-to-acoustic
compatibility boundary conditions. An acoustic problem is first solved with
acoustic AC3D8 elements, and pore pressure results should coincide with the
results of a combined AC3D8 plus C3D8A (with porosity close to 1) element model. An elastic problem is
first solved with solid C3D8 elements, and displacement and stress results should coincide
with the results of a combined C3D8 plus C3D8A (with zero porosity) element model.
Results and discussion
The results agree well with the results from the reference solutions.
- ac3d8_c3d8aconstr4.inp
- AC3D8 elements, 10 element mesh, to be compared against the results of
ac3d8_c3d8aconstr5.inp.
- ac3d8_c3d8aconstr5.inp
-
AC3D8 plus C3D8A elements, 15 element mesh, to be compared against the results of
ac3d8_c3d8aconstr4.inp.
- c3d8_c3d8aconstr3.inp
-
C3D8 elements, 10 element mesh, to be compared against the results of
c3d8_c3d8aconstr5.inp.
- c3d8_c3d8aconstr5.inp
-
C3D8 plus C3D8A elements, 15 element mesh, to be compared against the results of
c3d8_c3d8aconstr3.inp.
|