Newmark Time Integration Verification

Problem Statement and Results

In this one element test, the base of the 2D element is fixed in both x and y directions. The top of the block is also fixed in the y direction and a sawtooth acceleration is applied in the x direction.

The prescribed acceleration is integrated in time using Newmark time integration. For $var element = document.getElementById("moose-equation-b517f6e6-f2ef-4ea1-b917-e55623567b5c");katex.render("\\beta = 0.25", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-a984ae55-7efa-4e5a-88f2-78f7ed53d1fb");katex.render("\\gamma = 0.5", element, {displayMode:false,throwOnError:false});$ and timestep of 1.0 sec., the displacement and velocity are as given in Table 1. The interactive results shown in Figure 1 compare the analytical result ("previous") with the result computed from MASTODON for this problem ("current").

Table 1: Analytical displacement, acceleration, and velocity in the x-direction.

t	Displacement	Velocity	Acceleration
0	0	0	0
1	0.25	0.5	1
2	1	1	0
3	1.75	0.5	-1

Figure 1: Acceleration, velocity and displacement values computed using NewmarkBeta integration in MASTODON

Complete Input File

[Mesh<<<{"href": "../../../../syntax/Mesh/index.html"}>>>]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmin = 0.0
  xmax = 1.0
  ymin = 0.0
  ymax = 1.0
[]

[Variables<<<{"href": "../../../../syntax/Variables/index.html"}>>>]
  [./disp_x]
  [../]
  [./disp_y]
  [../]
[]

[AuxVariables<<<{"href": "../../../../syntax/AuxVariables/index.html"}>>>]
  [./vel_x]
  [../]
  [./accel_x]
  [../]
[]

[Kernels<<<{"href": "../../../../syntax/Kernels/index.html"}>>>]
  [./TensorMechanics<<<{"href": "../../../../syntax/Kernels/TensorMechanics/index.html"}>>>]
     displacements<<<{"description": "The nonlinear displacement variables for the problem"}>>> = 'disp_x disp_y'
  [../]
[]

[AuxKernels<<<{"href": "../../../../syntax/AuxKernels/index.html"}>>>]
  [./accel_x]
    type = NewmarkAccelAux<<<{"description": "Computes the current acceleration using the Newmark method.", "href": "../../../../source/auxkernels/NewmarkAccelAux.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = accel_x
    displacement<<<{"description": "displacement variable"}>>> = disp_x
    velocity<<<{"description": "velocity variable"}>>> = vel_x
    beta<<<{"description": "beta parameter for Newmark method"}>>> = 0.25
    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
  [../]
  [./vel_x]
    type = NewmarkVelAux<<<{"description": "Calculates the current velocity using Newmark method.", "href": "../../../../source/auxkernels/NewmarkVelAux.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = vel_x
    acceleration<<<{"description": "acceleration variable"}>>> = accel_x
    gamma<<<{"description": "gamma parameter for Newmark method"}>>> = 0.5
    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
  [../]
[]

[BCs<<<{"href": "../../../../syntax/BCs/index.html"}>>>]
  [./fixed_base_x]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    value<<<{"description": "Value of the BC"}>>> = 0.0
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_x
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = bottom
  [../]
  [./fixed_base_y]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../../source/bcs/DirichletBC.html"}>>>
    value<<<{"description": "Value of the BC"}>>> = 0.0
    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
  [../]
  [./top_x]
    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
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = top
    function<<<{"description": "Function describing the velocity."}>>> = acceleration_top
    beta<<<{"description": "beta parameter for Newmark time integration."}>>> = 0.25
    acceleration<<<{"description": "The acceleration variable."}>>> = accel_x
    velocity<<<{"description": "The velocity variable."}>>> = vel_x
  [../]
  [./top_y]
    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"}>>> = top
    value<<<{"description": "Value of the BC"}>>> = 0.0
  [../]
[]

[Functions<<<{"href": "../../../../syntax/Functions/index.html"}>>>]
  [./acceleration_top]
    type = PiecewiseLinear<<<{"description": "Linearly interpolates between pairs of x-y data", "href": "../../../../source/functions/PiecewiseLinear.html"}>>>
    x<<<{"description": "The abscissa values"}>>> = '0.0 1.0 2.0 3.0'
    y<<<{"description": "The ordinate values"}>>> = '0.0 1.0 0.0 -1.0'
  [../]
[../]

[Materials<<<{"href": "../../../../syntax/Materials/index.html"}>>>]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor<<<{"description": "Compute a constant isotropic elasticity tensor.", "href": "../../../../source/materials/ComputeIsotropicElasticityTensor.html"}>>>
    poissons_ratio<<<{"description": "Poisson's ratio for the material."}>>> = 0.3
    youngs_modulus<<<{"description": "Young's modulus of the material."}>>> = 1e9
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 0
  [../]
  [./strain]
    type = ComputeIncrementalSmallStrain<<<{"description": "Compute a strain increment and rotation increment for small strains.", "href": "../../../../source/materials/ComputeIncrementalStrain.html"}>>>
    displacements<<<{"description": "The displacements appropriate for the simulation geometry and coordinate system"}>>> = 'disp_x disp_y'
    block<<<{"description": "The list of blocks (ids or names) that this object will be applied"}>>> = 0
  [../]
  [./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"}>>> = 0
  [../]
[../]

[Executioner<<<{"href": "../../../../syntax/Executioner/index.html"}>>>]
  type = Transient
  start_time = 0.0
  end_time = 3.0
  dt = 1.0
[]

[Postprocessors<<<{"href": "../../../../syntax/Postprocessors/index.html"}>>>]
  [./disp_x]
    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."}>>> = '1.0 1.0 0.0'
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = disp_x
  [../]
  [./vel_x]
    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."}>>> = '1.0 1.0 0.0'
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = vel_x
  [../]
  [./accel_x]
    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."}>>> = '1.0 1.0 0.0'
    variable<<<{"description": "The name of the variable that this postprocessor operates on."}>>> = accel_x
  [../]
[]

[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
[]

(test/tests/kernels/time_integration/newmark.i)

# @requirement F1.1
#
# Newmark time integration test
#
# In this one element test, the base of the 2D element is fixed in both x and y
# directions. The top of the block is also fixed in the y direction and a sawtooth
# acceleration is applied in the x direction.

# The prescribed acceleration is integrated in time using newmark time integration.
# For beta = 0.25 and gamma = 0.5 and timestep of 1.0 s, the displacement and velocity
# are as given below.

#     t        disp_x     vel_x     accel_x
#     0          0          0         0
#     1          0.25       0.5       1
#     2          1          1         0
#     3          1.75       0.5      -1

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 1
  ny = 1
  xmin = 0.0
  xmax = 1.0
  ymin = 0.0
  ymax = 1.0
[]

[Variables]
  [./disp_x]
  [../]
  [./disp_y]
  [../]
[]

[AuxVariables]
  [./vel_x]
  [../]
  [./accel_x]
  [../]
[]

[Kernels]
  [./TensorMechanics]
     displacements = 'disp_x disp_y'
  [../]
[]

[AuxKernels]
  [./accel_x]
    type = NewmarkAccelAux
    variable = accel_x
    displacement = disp_x
    velocity = vel_x
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_x]
    type = NewmarkVelAux
    variable = vel_x
    acceleration = accel_x
    gamma = 0.5
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./fixed_base_x]
    type = DirichletBC
    value = 0.0
    variable = disp_x
    boundary = bottom
  [../]
  [./fixed_base_y]
    type = DirichletBC
    value = 0.0
    variable = disp_y
    boundary = bottom
  [../]
  [./top_x]
    type = PresetAcceleration
    variable = disp_x
    boundary = top
    function = acceleration_top
    beta = 0.25
    acceleration = accel_x
    velocity = vel_x
  [../]
  [./top_y]
    type = DirichletBC
    variable = disp_y
    boundary = top
    value = 0.0
  [../]
[]

[Functions]
  [./acceleration_top]
    type = PiecewiseLinear
    x = '0.0 1.0 2.0 3.0'
    y = '0.0 1.0 0.0 -1.0'
  [../]
[../]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    poissons_ratio = 0.3
    youngs_modulus = 1e9
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalSmallStrain
    displacements = 'disp_x disp_y'
    block = 0
  [../]
  [./stress]
    type = ComputeFiniteStrainElasticStress
    block = 0
  [../]
[../]

[Executioner]
  type = Transient
  start_time = 0.0
  end_time = 3.0
  dt = 1.0
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '1.0 1.0 0.0'
    variable = disp_x
  [../]
  [./vel_x]
    type = PointValue
    point = '1.0 1.0 0.0'
    variable = vel_x
  [../]
  [./accel_x]
    type = PointValue
    point = '1.0 1.0 0.0'
    variable = accel_x
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

Problem Statement and Results
Complete Input File

Usage

Development

Demonstration

Newmark Time Integration Verification

Problem Statement and Results

Complete Input File