3.1.10.35. FSIFluidBoundaryElement2D Element

3.1.10.35.1. Description

This command is used to construct an FSIFluidBoundaryElement2D element object. The FSIFluidBoundaryElement2D element is a 2-node linear acoustic boundary element object with the following features:

It is based on the Eulerian pressure formulation ([ZienkiewiczEtAl1978] , [ZienkiewiczEtAl2000] , [LøkkeEtAl2017] ) for (Class I) fluid-structure interaction problems.
It uses a 2 integration points Gauss quadrature.
Depending on the input variables, it enables the implementation of radiation boundary, reservoir bottom absorption or free surface effects.

3.1.10.35.2. Input Parameters

element FSIFluidBoundaryElement2D $eleTag $n1 $n2 $cc $alpha $g <-thickness $thickness>

Argument	Type	Description
$eleTag	integer	unique integer tag identifying element object
$n1 $n2	2 integers	the two nodes defining the element (-ndm 2 -ndf 1)
$cc	float	speed of pressure waves in water
$alpha	float	reservoir bottom reflection coefficient ([LøkkeEtAl2017])
$g	float	acceleration due to gravity
Optional:
$thickness	float	the thickness in 2D problems (default 1.0).

../../../../_images/FSIFluidBoundaryElement2D_geometry.png — Fig. 3.1.10.43 **Figure 1. Nodes, Gauss points and parent coordinate system**

3.1.10.35.3. Theory

For additional documentation regarding the derivation of the implemented finite elements (FSIFluidElement2D, FSIFluidBoundaryElement2D, FSIInterfaceElement2D) based on the Eulerian pressure formulation, please refer to the attached PDF document (Link to PDF)

3.1.10.35.4. Example

Three cases of valid inputs are shown below: 1. Radiation boundary, 2. Reservoir bottom absorption and 3. Surface waves effects.

Tcl Code

# set up a 2D-1DOF model: Radiation Boundary Side
model Basic -ndm 2 -ndf 1
node 11  0.0  0.0
node 22  1.0  1.0

# create the acoustic boundary element at radiation boundary side with speed of pressure waves in water, cc = 1.440000e+03 (set alpha = 0, g = 0 to exclude bottom absorption and mass terms for water surface, respectively)
set cc 1.440000e+03
element FSIFluidBoundaryElement2D 3  11 22  $cc 0.0 0.0 -thickness 1.0

    # set up a 2D-1DOF model: Reservoir Bottom Absorption Boundary
model Basic -ndm 2 -ndf 1
node 11  0.0  0.0
node 22  1.0  1.0

# create the acoustic boundary element at the bottom boundary of a reservoir given speed of pressure waves in water, cc = 1.440000e+03 and reservoir bottom reflection coefficient, alpha = 9.990000e-01 (set g = 0 to exclude mass terms for water surface)
set cc 1.440000e+03
    set alpha 9.990000e-01
element FSIFluidBoundaryElement2D 4  11 22  $cc $alpha 0.0 -thickness 1.0

    # set up a 2D-1DOF model: Free Surface Boundary
model Basic -ndm 2 -ndf 1
node 11  0.0  0.0
node 22  1.0  1.0

# create the acoustic boundary element at the free surface boundary of the reservoir given g = 9.807 (set alpha = 0, cc = 0 to exclude damping terms)
set cc 1.440000e+03
    set alpha 9.990000e-01
set g 9.807
    element FSIFluidBoundaryElement2D 5  11 22   0.0 0.0 $g -thickness 1.0

Python Code

# set up a 2D-1DOF model: Radiation Boundary Side
model('Basic', '-ndm', 2, '-ndf', 1)
node(11, 0.0, 0.0)
node(22, 1.0, 1.0)

# create the acoustic boundary element at radiation boundary side
cc = 1.440000e+03
element('FSIFluidBoundaryElement2D', 3, 11, 22, cc, 0.0, 0.0, thickness=1.0)

    # set up a 2D-1DOF model: Reservoir Bottom Absorption Boundary
model('Basic', '-ndm', 2, '-ndf', 1)
node(11, 0.0, 0.0)
node(22, 1.0, 1.0)

# create the acoustic boundary element at the bottom boundary of a reservoir
cc = 1.440000e+03
    alpha = 9.990000e-01
element('FSIFluidBoundaryElement2D', 4, 11, 22, cc, alpha, 0.0, thickness=1.0)

    # set up a 2D-1DOF model: Free Surface Boundary
model('Basic', '-ndm', 2, '-ndf', 1)
node(11, 0.0, 0.0)
node(22, 1.0, 1.0)

# create the acoustic boundary element at the free surface boundary of the reservoir
g = 9.807
element('FSIFluidBoundaryElement2D', 5, 11, 22, 0.0, 0.0, g, thickness=1.0)

Code Developed, implemented and tested by:

Massimo Petracca (ASDEA Software),
Enrique Simbort (UC San Diego),
Joel Conte (UC San Diego).

3.1.10.35.5. References

[ZienkiewiczEtAl1978]

Zienkiewicz O.C., Bettess P. (1978) “Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment”, Inter. J. Numer. Meth. Eng.., 13(1): 1–16. (Link to article)

[ZienkiewiczEtAl2000]

Zienkiewicz O.C., Taylor R.L. (2000) “The Finite Element Method”, Butterworth-Heinemann, Vol.1, 5th Ed., Ch.19.

[LøkkeEtAl2017] (1,2)

Løkke A., Chopra A.K. (2017) “Direct finite element method for nonlinear analysis of semi-unbounded dam–water–foundation rock systems”, Earthquake Engineering and Structural Dynamics 46(8): 1267–1285. (Link to article)