Example 12a: Fluid-structure interaction analysis of a tank subjected to uni-directional harmonic acceleration

note:Units of this example

Giga kilograms, Giga Newtons, meters, and seconds were used for the modeling. However, the post-processed results maybe presented in different units.

Model description

This example demonstrates the fluid-structure interaction between a steel tank and the fluid inside it. The tank is cylindrical with height 2 $var element = document.getElementById("moose-equation-ae092adb-e4bd-4f93-9f70-0712b1f661bd");katex.render("m", element, {displayMode:false,throwOnError:false});$ , thickness 7.9 $var element = document.getElementById("moose-equation-40013439-a99d-4e9d-ada4-cb74ee2f1165");katex.render("mm", element, {displayMode:false,throwOnError:false});$ , and radius 0.79 $var element = document.getElementById("moose-equation-80c11df4-e920-4a5d-b595-8171be2e0ee9");katex.render("m", element, {displayMode:false,throwOnError:false});$ . The fluid inside the tank is water with height 1.8 $var element = document.getElementById("moose-equation-d507fd9a-14f0-4ad1-9480-8ad5281f59e9");katex.render("m", element, {displayMode:false,throwOnError:false});$ . Steel has an elastic modulus of 200 $var element = document.getElementById("moose-equation-90e07eeb-b5f7-4999-9751-5787ed2c1176");katex.render("GPa", element, {displayMode:false,throwOnError:false});$ and a Poisson's ratio of 0.27. Water has a density of 1E-6 $var element = document.getElementById("moose-equation-d2b71eda-561a-42fc-9b80-bd386c11e027");katex.render("Giga~kg/m^3", element, {displayMode:false,throwOnError:false});$ and a speed of sound of 1466.47 $var element = document.getElementById("moose-equation-71a4659a-fc68-4108-b416-0e4f85a14436");katex.render("m/s", element, {displayMode:false,throwOnError:false});$ . This tank with water is subjected to a harmonic acceleration in the x-direction. Figure 1 presents a schematic of the tank and its dimensions.

Figure 1: Schematic of the tank and its dimensions.

Input motion

The bottom of the tank is subjected to two cycles of sinusoidal acceleration with peak amplitude of 0.025 $var element = document.getElementById("moose-equation-4f9f9f9e-ec2c-4883-af39-01d6ae1f5236");katex.render("g", element, {displayMode:false,throwOnError:false});$ . The length of the input motion is about 4 $var element = document.getElementById("moose-equation-971f9bca-f007-484d-bf5b-929ef7baeac7");katex.render("s", element, {displayMode:false,throwOnError:false});$ . Figure 2 presents a plot of the input motion.

Figure 2: Input harmonic motion to the tank.

Finite element mesh generation

The three-dimensional finite element mesh is generated using Cubit 15.5b. The following Cubit input file presents the script to generate the mesh. Figure 3 presents the finite-element mesh used for this example.

reset
create cylinder radius 0.7979 height 2.0
volume 1 move z 1.0
create cylinder radius 0.79 height 2.1
volume 2 move z 1.0579
subtract volume 2 from volume 1
create cylinder radius 0.79 height 1.8
volume 3 move z 0.9079
merge all
webcut volume 1  with sheet extended from surface 7 
webcut volume 1  with sheet extended from surface 8 
webcut volume 5  with sheet extended from surface 12 
merge all 
merge surface 26 with surface 10 force
merge surface 14 with surface 17 force

surface 9 scheme hole rad_intervals 2
surface 9 size 0.1
mesh surface 9

volume 5 int 3
mesh volume 5

volume 6 int 18
mesh volume 6
mesh volume 1

mesh volume 4

volume 3 int 18
mesh volume 3

block 1 add volume 6 4 1 5
block 2 add volume 3

sideset 1 add surface 15 13
sideset 1 name "Bottom"

sideset 4 add surface 12
sideset 4 name "Fluid_top"

Figure 3: Finite-element mesh of the tank.

Modeling using MOOSE modules

The tank with water is modeled using the capabilities of the Fluid-Structure Interaction Module and the Solid Mechanics Module. In particular, the tank is modeled using the following kernels in the Solid Mechanics Module:

InertialForce kernel for the acceleration component
DynamicTensorMechanics action for the stiffness component

The fluid is modeled using the following kernels in the Fluid-Structure Interaction Module:

AcousticInertia kernel for the double time derivative of the fluid pressure
Diffusion kernel for the Laplacian of the fluid pressure

Fluid-structure interaction between the tank and the fluid is modeled using the StructureAcousticInterface interface kernel. This kernel enforces displacement and stress/pressure continuity between the tank and the fluid. The input file for executing the model is presented below.

[Mesh<<<{"href": "../syntax/Mesh/index.html"}>>>]
  [file]
    type = FileMeshGenerator<<<{"description": "Read a mesh from a file.", "href": "../source/meshgenerators/FileMeshGenerator.html"}>>>
    file<<<{"description": "The filename to read."}>>> = Model_1.e
  []
  [interface1]
    type = SideSetsBetweenSubdomainsGenerator<<<{"description": "MeshGenerator that creates a sideset composed of the nodes located between two or more subdomains.", "href": "../source/meshgenerators/SideSetsBetweenSubdomainsGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = file
    primary_block<<<{"description": "The primary set of blocks for which to draw a sideset between"}>>> = '2'
    paired_block<<<{"description": "The paired set of blocks for which to draw a sideset between"}>>> = '1'
    new_boundary<<<{"description": "The list of boundary names to create on the supplied subdomain"}>>> = 'Interface'
  []
[]

[GlobalParams<<<{"href": "../syntax/GlobalParams/index.html"}>>>]
[]

[Variables<<<{"href": "../syntax/Variables/index.html"}>>>]
  [p]
   block = 2
  []
  [disp_x]
    block = 1
  []
  [disp_y]
    block = 1
  []
  [disp_z]
    block = 1
  []
[]

[AuxVariables<<<{"href": "../syntax/AuxVariables/index.html"}>>>]
  [Wave1]
    block = 2
  []
  [vel_x]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [accel_x]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [vel_y]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [accel_y]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [vel_z]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [accel_z]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
    block = 1
  []
  [stress_xx]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
  [stress_yy]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
  [stress_xy]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
  [stress_zz]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
  [stress_yz]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
  [stress_xz]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
    block = 1
  []
[]

[Kernels<<<{"href": "../syntax/Kernels/index.html"}>>>]
  [diffusion]
    type = Diffusion<<<{"description": "The Laplacian operator ($-\\nabla \\cdot \\nabla u$), with the weak form of $(\\nabla \\phi_i, \\nabla u_h)$.", "href": "../source/kernels/Diffusion.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = 'p'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [inertia]
    type = AcousticInertia<<<{"description": "Calculates the residual for the inertial force which is the double time derivative of pressure.", "href": "../source/kernels/AcousticInertia.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [DynamicTensorMechanics<<<{"href": "../syntax/Kernels/DynamicTensorMechanics/index.html"}>>>]
    displacements<<<{"description": "The nonlinear displacement variables for the problem"}>>> = 'disp_x disp_y disp_z'
    block<<<{"description": "The list of ids of the blocks (subdomain) that the stress divergence kernels will be applied to"}>>> = 1
  []
  [inertia_x1]
    type = InertialForce<<<{"description": "Calculates the residual for the inertial force ($M \\cdot acceleration$) and the contribution of mass dependent Rayleigh damping and HHT time  integration scheme ($\\eta \\cdot M \\cdot ((1+\\alpha)velq2-\\alpha \\cdot vel-old) $)", "href": "../source/kernels/InertialForce.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_x
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [inertia_y1]
    type = InertialForce<<<{"description": "Calculates the residual for the inertial force ($M \\cdot acceleration$) and the contribution of mass dependent Rayleigh damping and HHT time  integration scheme ($\\eta \\cdot M \\cdot ((1+\\alpha)velq2-\\alpha \\cdot vel-old) $)", "href": "../source/kernels/InertialForce.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [inertia_z1]
    type = InertialForce<<<{"description": "Calculates the residual for the inertial force ($M \\cdot acceleration$) and the contribution of mass dependent Rayleigh damping and HHT time  integration scheme ($\\eta \\cdot M \\cdot ((1+\\alpha)velq2-\\alpha \\cdot vel-old) $)", "href": "../source/kernels/InertialForce.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_z
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
[]

[AuxKernels<<<{"href": "../syntax/AuxKernels/index.html"}>>>]
  [waves]
    type = WaveHeightAuxKernel<<<{"description": "Calculates the wave heights given pressures.", "href": "../source/auxkernels/WaveHeightAuxKernel.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = 'Wave1'
    pressure<<<{"description": "pressure variable"}>>> = p
    density<<<{"description": "fluid density"}>>> = 1e-6
    gravity<<<{"description": "acceleration due to gravity"}>>> = 9.81
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [accel_x]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_x
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = accel_x
    first<<<{"description": "Set to true to copy the first derivative to the auxvariable. If false, the second derivative is copied."}>>> = false
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [vel_x]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_x
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = vel_x
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [accel_y]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_y
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = accel_y
    first<<<{"description": "Set to true to copy the first derivative to the auxvariable. If false, the second derivative is copied."}>>> = false
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [vel_y]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_y
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = vel_y
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [accel_z]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_z
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = accel_z
    first<<<{"description": "Set to true to copy the first derivative to the auxvariable. If false, the second derivative is copied."}>>> = false
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [vel_z]
    type = TestNewmarkTI<<<{"description": "Assigns the velocity/acceleration calculated by time integrator to the velocity/acceleration auxvariable.", "href": "../source/auxkernels/TestNewmarkTI.html"}>>>
    displacement<<<{"description": "The variable whose first/second derivative needs to be copied to the provided auxvariable."}>>> = disp_z
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = vel_z
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_xx]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_xx
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 0
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 0
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_yy]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_yy
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 1
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 1
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_xy]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_xy
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 0
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 1
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_zz]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_zz
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 2
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 2
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_yz]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_yz
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 1
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 2
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [stress_xz]
    type = RankTwoAux<<<{"description": "Access a component of a RankTwoTensor", "href": "../source/auxkernels/RankTwoAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = stress
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = stress_xz
    index_i<<<{"description": "The index i of ij for the tensor to output (0, 1, 2)"}>>> = 0
    index_j<<<{"description": "The index j of ij for the tensor to output (0, 1, 2)"}>>> = 2
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
[]

[InterfaceKernels<<<{"href": "../syntax/InterfaceKernels/index.html"}>>>]
  [interface1]
    type = StructureAcousticInterface<<<{"description": "Enforces displacement and stress/pressure continuity between the fluid and structural domains. Element is always the structure and neighbor is always the fluid.", "href": "../source/interfacekernels/StructureAcousticInterface.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = disp_x
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = 'Interface'
    D<<<{"description": "Fluid density."}>>> = 1e-6
    component<<<{"description": "The desired component of displacement."}>>> = 0
  []
  [interface2]
    type = StructureAcousticInterface<<<{"description": "Enforces displacement and stress/pressure continuity between the fluid and structural domains. Element is always the structure and neighbor is always the fluid.", "href": "../source/interfacekernels/StructureAcousticInterface.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = disp_y
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = 'Interface'
    D<<<{"description": "Fluid density."}>>> = 1e-6
    component<<<{"description": "The desired component of displacement."}>>> = 1
  []
  [interface3]
    type = StructureAcousticInterface<<<{"description": "Enforces displacement and stress/pressure continuity between the fluid and structural domains. Element is always the structure and neighbor is always the fluid.", "href": "../source/interfacekernels/StructureAcousticInterface.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p
    neighbor_var<<<{"description": "The variable on the other side of the interface."}>>> = disp_z
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies"}>>> = 'Interface'
    D<<<{"description": "Fluid density."}>>> = 1e-6
    component<<<{"description": "The desired component of displacement."}>>> = 2
  []
[]

[BCs<<<{"href": "../syntax/BCs/index.html"}>>>]
  [bottom_accel1]
    type = PresetAcceleration<<<{"description": "Prescribe acceleration on a given boundary in a given direction", "href": "../source/bcs/PresetAcceleration.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_x
    velocity<<<{"description": "The velocity variable."}>>> = vel_x
    acceleration<<<{"description": "The acceleration variable."}>>> = accel_x
    beta<<<{"description": "beta parameter for Newmark time integration."}>>> = 0.25
    function<<<{"description": "Function describing the velocity."}>>> = accel_bottom
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'Bottom'
  []
  [disp_x2]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_y
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'Bottom'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []
  [disp_x3]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_z
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'Bottom'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []
  [free]
    type = FluidFreeSurfaceBC<<<{"description": "Applies a mixed Dirichlet-Neumann BC on the fluid surface.", "href": "../source/bcs/FluidFreeSurfaceBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = p
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'Fluid_top'
    alpha<<<{"description": "Inverse of the acceleration due to gravity."}>>> = '0.1'
  []
[]

[Functions<<<{"href": "../syntax/Functions/index.html"}>>>]
  [accel_bottom]
    type = PiecewiseLinear<<<{"description": "Linearly interpolates between pairs of x-y data", "href": "../source/functions/PiecewiseLinear.html"}>>>
    data_file<<<{"description": "File holding CSV data"}>>> = SC_harmonic_input.csv
    scale_factor<<<{"description": "Scale factor to be applied to the output values"}>>> = 9.81
    format<<<{"description": "Format of csv data file that is in either in columns or rows"}>>> = 'columns'
  []
[]

[Materials<<<{"href": "../syntax/Materials/index.html"}>>>]
  [density]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../source/materials/GenericConstantMaterial.html"}>>>
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = inv_co_sq
    prop_values<<<{"description": "The values associated with the named properties"}>>> = 4.65e-7
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 2
  []
  [density0]
    type = GenericConstantMaterial<<<{"description": "Declares material properties based on names and values prescribed by input parameters.", "href": "../source/materials/GenericConstantMaterial.html"}>>>
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
    prop_names<<<{"description": "The names of the properties this material will have"}>>> = density
    prop_values<<<{"description": "The values associated with the named properties"}>>> = 7.85e-6
  []
  [elasticity_base]
    type = ComputeIsotropicElasticityTensor<<<{"description": "Compute a constant isotropic elasticity tensor.", "href": "../source/materials/ComputeIsotropicElasticityTensor.html"}>>>
    youngs_modulus<<<{"description": "Young's modulus of the material."}>>> = 2e2
    poissons_ratio<<<{"description": "Poisson's ratio for the material."}>>> = 0.27
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
  [strain]
    type = ComputeFiniteStrain<<<{"description": "Compute a strain increment and rotation increment for finite strains.", "href": "../source/materials/ComputeFiniteStrain.html"}>>>
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
    displacements<<<{"description": "The displacements appropriate for the simulation geometry and coordinate system"}>>> = 'disp_x disp_y disp_z'
  []
  [stress]
    type =  ComputeFiniteStrainElasticStress<<<{"description": "Compute stress using elasticity for finite strains", "href": "../source/materials/ComputeFiniteStrainElasticStress.html"}>>>
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 1
  []
[]

[Preconditioning<<<{"href": "../syntax/Preconditioning/index.html"}>>>]
  [andy]
    type = SMP<<<{"description": "Single matrix preconditioner (SMP) builds a preconditioner using user defined off-diagonal parts of the Jacobian.", "href": "../source/preconditioners/SingleMatrixPreconditioner.html"}>>>
    full<<<{"description": "Set to true if you want the full set of couplings between variables simply for convenience so you don't have to set every off_diag_row and off_diag_column combination."}>>> = true
  []
[]

[Executioner<<<{"href": "../syntax/Executioner/index.html"}>>>]
  type = Transient
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'
  start_time = 0.0
  end_time = 4.0
  dt = 0.005
  dtmin = 0.00001
  nl_abs_tol = 1e-14
  nl_rel_tol = 1e-14
  l_tol = 1e-14
  l_max_its = 25
  timestep_tolerance = 1e-8
  automatic_scaling = true
  [TimeIntegrator<<<{"href": "../syntax/Executioner/TimeIntegrator/index.html"}>>>]
    type = NewmarkBeta
  []
[]

[Postprocessors<<<{"href": "../syntax/Postprocessors/index.html"}>>>]
  [PBottom]
    type = PointValue<<<{"description": "Compute the value of a variable at a specified location", "href": "../source/postprocessors/PointValue.html"}>>>
    point<<<{"description": "The physical point where the solution will be evaluated."}>>> = '0.79 0.0 0.0079'
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = p
  []
  [Wave]
    type = PointValue<<<{"description": "Compute the value of a variable at a specified location", "href": "../source/postprocessors/PointValue.html"}>>>
    point<<<{"description": "The physical point where the solution will be evaluated."}>>> = '0.71 0.0 1.8079'
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = Wave1
  []
[]

[Outputs<<<{"href": "../syntax/Mastodon/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  exodus<<<{"description": "Output the results using the default settings for Exodus output."}>>> = true
  perf_graph<<<{"description": "Enable printing of the performance graph to the screen (Console)"}>>> = true
  print_linear_residuals<<<{"description": "Enable printing of linear residuals to the screen (Console)"}>>> = true
  file_base<<<{"description": "Common file base name to be utilized with all output objects"}>>> = Ex_SC_Harmonic
  [out]
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'TIMESTEP_BEGIN'
    type = CSV<<<{"description": "Output for postprocessors, vector postprocessors, and scalar variables using comma seperated values (CSV).", "href": "../source/outputs/CSV.html"}>>>
    file_base<<<{"description": "The desired solution output name without an extension. If not provided, MOOSE sets it with Outputs/file_base when available. Otherwise, MOOSE uses input file name and this object name for a master input or uses master file_base, the subapp name and this object name for a subapp input to set it."}>>> = SC_Harmonic
  []
[]

Results: Pressures on the tank and wave heights over the fluid

Pressures acting on the tank wall as a function of time are recorded at the circular red dot as illustrated in Figure 1. Wave heights as the function of time are also recorded at the triangular red dot as illustrated in Figure 1. Figure 5 and Figure 6 present the pressures and wave heights, respectively.

Figure 5: Pressures recorded at the circular dot as illustrated in Figure 1.

Figure 6: Wave heights recorded at the triangular dot as illustrated in Figure 1.

Results: Comparison to a reference solution

Figure 4 presents the peak pressure and peak wave height in comparison to a reference solution from Yu and Whittaker (2020). It is observed that the modeled results match quite well with the reference solution.

Figure 4: Comparison of the modeled results to a reference solution from Yu and Whittaker (2020).

References

C. C. Yu and A. S. Whittaker. Analytical solutions for seismic fluid-structure interaction of head-supported cylindrical tanks. Journal of Engineering Mechanics, 146(10):04020112, 2020.

@article{Yu2020a,
    author = "Yu, C. C. and Whittaker, A. S.",
    journal = "Journal of Engineering Mechanics",
    title = "{Analytical solutions for seismic fluid-structure interaction of head-supported cylindrical tanks}",
    volume = "146",
    number = "10",
    year = "2020",
    pages = "04020112"
}

(examples/ex12a/Model_1.jou)

reset
create cylinder radius 0.7979 height 2.0
volume 1 move z 1.0
create cylinder radius 0.79 height 2.1
volume 2 move z 1.0579
subtract volume 2 from volume 1
create cylinder radius 0.79 height 1.8
volume 3 move z 0.9079
merge all
webcut volume 1  with sheet extended from surface 7 
webcut volume 1  with sheet extended from surface 8 
webcut volume 5  with sheet extended from surface 12 
merge all 
merge surface 26 with surface 10 force
merge surface 14 with surface 17 force

surface 9 scheme hole rad_intervals 2
surface 9 size 0.1
mesh surface 9

volume 5 int 3
mesh volume 5

volume 6 int 18
mesh volume 6
mesh volume 1

mesh volume 4

volume 3 int 18
mesh volume 3


block 1 add volume 6 4 1 5
block 2 add volume 3

sideset 1 add surface 15 13
sideset 1 name "Bottom"

sideset 4 add surface 12
sideset 4 name "Fluid_top"

(examples/ex12a/FSI_ex12a.i)

[Mesh]
  [file]
    type = FileMeshGenerator
    file = Model_1.e
  []
  [interface1]
    type = SideSetsBetweenSubdomainsGenerator
    input = file
    primary_block = '2'
    paired_block = '1'
    new_boundary = 'Interface'
  []
[]

[GlobalParams]
[]

[Variables]
  [p]
   block = 2
  []
  [disp_x]
    block = 1
  []
  [disp_y]
    block = 1
  []
  [disp_z]
    block = 1
  []
[]

[AuxVariables]
  [Wave1]
    block = 2
  []
  [vel_x]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [accel_x]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [vel_y]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [accel_y]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [vel_z]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [accel_z]
    order = FIRST
    family = LAGRANGE
    block = 1
  []
  [stress_xx]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
  [stress_yy]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
  [stress_xy]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
  [stress_zz]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
  [stress_yz]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
  [stress_xz]
    order = CONSTANT
    family = MONOMIAL
    block = 1
  []
[]

[Kernels]
  [diffusion]
    type = Diffusion
    variable = 'p'
    block = 2
  []
  [inertia]
    type = AcousticInertia
    variable = p
    block = 2
  []
  [DynamicTensorMechanics]
    displacements = 'disp_x disp_y disp_z'
    block = 1
  []
  [inertia_x1]
    type = InertialForce
    variable = disp_x
    block = 1
  []
  [inertia_y1]
    type = InertialForce
    variable = disp_y
    block = 1
  []
  [inertia_z1]
    type = InertialForce
    variable = disp_z
    block = 1
  []
[]

[AuxKernels]
  [waves]
    type = WaveHeightAuxKernel
    variable = 'Wave1'
    pressure = p
    density = 1e-6
    gravity = 9.81
    execute_on = timestep_end
    block = 2
  []
  [accel_x]
    type = TestNewmarkTI
    displacement = disp_x
    variable = accel_x
    first = false
    block = 1
  []
  [vel_x]
    type = TestNewmarkTI
    displacement = disp_x
    variable = vel_x
    block = 1
  []
  [accel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = accel_y
    first = false
    block = 1
  []
  [vel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = vel_y
    block = 1
  []
  [accel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = accel_z
    first = false
    block = 1
  []
  [vel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = vel_z
    block = 1
  []
  [stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_i = 0
    index_j = 0
    block = 1
  []
  [stress_yy]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_yy
    index_i = 1
    index_j = 1
    block = 1
  []
  [stress_xy]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xy
    index_i = 0
    index_j = 1
    block = 1
  []
  [stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_i = 2
    index_j = 2
    block = 1
  []
  [stress_yz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_yz
    index_i = 1
    index_j = 2
    block = 1
  []
  [stress_xz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xz
    index_i = 0
    index_j = 2
    block = 1
  []
[]

[InterfaceKernels]
  [interface1]
    type = StructureAcousticInterface
    variable = p
    neighbor_var = disp_x
    boundary = 'Interface'
    D = 1e-6
    component = 0
  []
  [interface2]
    type = StructureAcousticInterface
    variable = p
    neighbor_var = disp_y
    boundary = 'Interface'
    D = 1e-6
    component = 1
  []
  [interface3]
    type = StructureAcousticInterface
    variable = p
    neighbor_var = disp_z
    boundary = 'Interface'
    D = 1e-6
    component = 2
  []
[]

[BCs]
  [bottom_accel1]
    type = PresetAcceleration
    variable = disp_x
    velocity = vel_x
    acceleration = accel_x
    beta = 0.25
    function = accel_bottom
    boundary = 'Bottom'
  []
  [disp_x2]
    type = DirichletBC
    variable = disp_y
    boundary = 'Bottom'
    value = 0.0
  []
  [disp_x3]
    type = DirichletBC
    variable = disp_z
    boundary = 'Bottom'
    value = 0.0
  []
  [free]
    type = FluidFreeSurfaceBC
    variable = p
    boundary = 'Fluid_top'
    alpha = '0.1'
  []
[]

[Functions]
  [accel_bottom]
    type = PiecewiseLinear
    data_file = SC_harmonic_input.csv
    scale_factor = 9.81
    format = 'columns'
  []
[]

[Materials]
  [density]
    type = GenericConstantMaterial
    prop_names = inv_co_sq
    prop_values = 4.65e-7
    block = 2
  []
  [density0]
    type = GenericConstantMaterial
    block = 1
    prop_names = density
    prop_values = 7.85e-6
  []
  [elasticity_base]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 2e2
    poissons_ratio = 0.27
    block = 1
  []
  [strain]
    type = ComputeFiniteStrain
    block = 1
    displacements = 'disp_x disp_y disp_z'
  []
  [stress]
    type =  ComputeFiniteStrainElasticStress
    block = 1
  []
[]

[Preconditioning]
  [andy]
    type = SMP
    full = true
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'
  start_time = 0.0
  end_time = 4.0
  dt = 0.005
  dtmin = 0.00001
  nl_abs_tol = 1e-14
  nl_rel_tol = 1e-14
  l_tol = 1e-14
  l_max_its = 25
  timestep_tolerance = 1e-8
  automatic_scaling = true
  [TimeIntegrator]
    type = NewmarkBeta
  []
[]

[Postprocessors]
  [PBottom]
    type = PointValue
    point = '0.79 0.0 0.0079'
    variable = p
  []
  [Wave]
    type = PointValue
    point = '0.71 0.0 1.8079'
    variable = Wave1
  []
[]

[Outputs]
  csv = true
  exodus = true
  perf_graph = true
  print_linear_residuals = true
  file_base = Ex_SC_Harmonic
  [out]
    execute_on = 'TIMESTEP_BEGIN'
    type = CSV
    file_base = SC_Harmonic
  []
[]

Model description
Input motion
Finite element mesh generation
Modeling using MOOSE modules
Results : Pressures on the tank and wave heights over the fluid
Results : Comparison to a reference solution
References

Usage

Development

Demonstration