NewmarkBeta

Computes the first and second time derivative of variable using Newmark-Beta method.

Description

Newmark time integration (Newmark, 1959) is one of the commonly used time integration methods in structural dynamics problems. In this method, the second ( $var element = document.getElementById("moose-equation-5a796c72-3a95-49b7-b8bf-c48cb610a979");katex.render("\\ddot{u}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) and first ( $var element = document.getElementById("moose-equation-92053ef0-c173-4aa7-8a8f-db47a20db38d");katex.render("\\dot{u}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) time derivatives of a variable $var element = document.getElementById("moose-equation-dcbe5c03-1084-4aeb-a2f8-f2e75eb2f21d");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ at $var element = document.getElementById("moose-equation-f46e32db-c249-488a-8df1-83966b3feccd");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are written in terms of the $var element = document.getElementById("moose-equation-1510295b-44ff-40bc-b870-f0d66da0d025");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-874eeaae-ea1d-4bc8-a81c-7418e910206b");katex.render("\\dot{u}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-bbf5528a-bf8e-47d3-a301-96b9c7cd93a4");katex.render("\\ddot{u}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ at time $var element = document.getElementById("moose-equation-8b424f2a-3717-490f-88e5-7a9a9873feef");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , and $var element = document.getElementById("moose-equation-09ac0a14-2336-45ba-82f5-c935f7549b8e");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ at $var element = document.getElementById("moose-equation-a60c5f78-efc7-42c1-bd84-e23956274e92");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ as shown below:

$var element = document.getElementById("moose-equation-c660b5dd-d6ed-4480-bc38-b498d35145d0");katex.render("\\begin{aligned} \\mathbf{\\ddot{u}}(t+\\Delta t) &=& \\frac{\\mathbf{u}(t+\\Delta t)-\\mathbf{u}(t)}{\\beta \\Delta t^2}- \\frac{\\mathbf{\\dot{u}}(t)}{\\beta \\Delta t}+\\frac{\\beta -0.5}{\\beta}\\mathbf{\\ddot{u}}(t) \\\\ \\mathbf{\\dot{u}}(t+ \\Delta t) &=& \\mathbf{\\dot{u}}(t)+ (1-\\gamma)\\Delta t \\mathbf{\\ddot{u}}(t) + \\gamma \\Delta t \\mathbf{\\ddot{u}}(t+\\Delta t) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

In the above equations, $var element = document.getElementById("moose-equation-760d6723-b13a-4f39-862c-27dde2fa9397");katex.render("\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-525d60d2-e22c-4ef0-b4cb-ed02ccfff510");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are Newmark time integration parameters.

For $var element = document.getElementById("moose-equation-8a6ebdda-cbe1-41c7-b7a8-adfda3daf5ef");katex.render("\\beta = \\frac{1}{4}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-bd4dae09-3473-46ab-aab3-39f9623989b5");katex.render("\\gamma = \\frac{1}{2}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , the Newmark time integration method is implicit, unconditionally stable and second order accurate in time. This is the constant average acceleration method with no numerical damping.
$var element = document.getElementById("moose-equation-0564bb8d-7969-4623-90aa-f117716d662b");katex.render("\\beta = \\frac{1}{6}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-02ea5497-3985-4266-95d9-6fc2c1415d0f");katex.render("\\gamma = \\frac{1}{2}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ results in the linear acceleration method where the acceleration is linearly varying between $var element = document.getElementById("moose-equation-27222a4b-1efd-43fa-8b9d-53679b7dcbae");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-c19e7dd4-6779-4ea6-b389-c8af1c42223b");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . This method is also implicit, unconditionally stable and second order accurate in time. However, there is a small numerical damping when the linear acceleration method is used.
For $var element = document.getElementById("moose-equation-b1932b99-45b8-4885-bf61-7c42710d18d6");katex.render("\\gamma = \\frac{1}{2}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , the method is second order accurate and it is unconditionally stable for $var element = document.getElementById("moose-equation-b2ebfa60-abb3-4790-bf5d-000f73954af5");katex.render("\\frac{1}{2} \\le \\gamma \\le 2 \\beta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

When using the constant average acceleration method that has no numerical damping, high frequency noise can sometimes be observed in the velocity and acceleration time histories for a problem with prescribed displacement (Bathe and Noh, 2012). Using other parameters for $var element = document.getElementById("moose-equation-c98bd2e0-8feb-4e83-b4c6-c352ad9e07f7");katex.render("\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-2c1ed5fc-412f-4966-a02d-82bf1808b804");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ results in non-zero numerical damping that damps out part of the high frequency noise but not all of it. Hilber-Hughes-Taylor (HHT) time integration is a variation of the Newmark method that damps out high frequency noise especially in structural dynamics problems. More details about this Newmark and HHT time integration schemes can be found in these lecture notes. HHT time integration requires modification to the equation of motion and is currently implemented only for structural dynamics problems in tensor mechanics module.

Input Parameters

beta0.25beta value
Default:0.25
C++ Type:double
Options:
Description:beta value
gamma0.5gamma value
Default:0.5
C++ Type:double
Options:
Description:gamma value

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

modules/tensor_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht_ti.i
modules/tensor_mechanics/test/tests/dynamics/wave_1D/wave_rayleigh_hht_ti.i
test/tests/kernels/vector_dot_dot/vector_test.i
test/tests/variables/time_derivatives_neighbor/test.i
test/tests/time_integrators/newmark-beta/newmark_beta_prescribed_parameters.i

modules/tensor_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht_ti.i

# Test for damped small strain euler beam vibration in y direction

# An impulse load is applied at the end of a cantilever beam of length 4m.
# The properties of the cantilever beam are as follows:
# Young's modulus (E) = 1e4
# Shear modulus (G) = 4e7
# Shear coefficient (k) = 1.0
# Cross-section area (A) = 0.01
# Iy = 1e-4 = Iz
# Length (L)= 4 m
# density (rho) = 1.0
# mass proportional rayleigh damping(eta) = 0.1
# stiffness proportional rayleigh damping(eta) = 0.1
# HHT time integration parameter (alpha) = -0.3
# Corresponding Newmark beta time integration parameters beta = 0.4225 and gamma = 0.8

# For this beam, the dimensionless parameter alpha = kAGL^2/EI = 6.4e6
# Therefore, the behaves like a Euler-Bernoulli beam.

# The displacement time history from this analysis matches with that obtained from Abaqus.

# Values from the first few time steps are as follows:
# time  disp_y                vel_y                accel_y
# 0.0   0.0                   0.0                  0.0
# 0.2   0.019898364318588     0.18838688112273     1.1774180070171
# 0.4   0.045577003505278     0.087329917525455   -0.92596052423724
# 0.6   0.063767907208218     0.084330765885995    0.21274543331268
# 0.8   0.073602908614573     0.020029576220975   -0.45506879373455
# 1.0   0.06841704414745     -0.071840076837194   -0.46041813317992

[Mesh]
  type = GeneratedMesh
  nx = 10
  dim = 1
  xmin = 0.0
  xmax = 4.0
  displacements = 'disp_x disp_y disp_z'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_z]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./accel_x] # These auxkernels are only to check output
    type = TestNewmarkTI
    displacement = disp_x
    variable = accel_x
    first = false
  [../]
  [./accel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = accel_y
    first = false
  [../]
  [./accel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = accel_z
    first = false
  [../]
  [./vel_x]
    type = TestNewmarkTI
    displacement = disp_x
    variable = vel_x
  [../]
  [./vel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = vel_y
  [../]
  [./vel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = vel_z
  [../]
  [./rot_accel_x]
    type = TestNewmarkTI
    displacement = rot_x
    variable = rot_accel_x
    first = false
  [../]
  [./rot_accel_y]
    type = TestNewmarkTI
    displacement = rot_y
    variable = rot_accel_y
    first = false
  [../]
  [./rot_accel_z]
    type = TestNewmarkTI
    displacement = rot_z
    variable = rot_accel_z
    first = false
  [../]
  [./rot_vel_x]
    type = TestNewmarkTI
    displacement = rot_x
    variable = rot_vel_x
  [../]
  [./rot_vel_y]
    type = TestNewmarkTI
    displacement = rot_y
    variable = rot_vel_y
  [../]
  [./rot_vel_z]
    type = TestNewmarkTI
    displacement = rot_z
    variable = rot_vel_z
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./fixy1]
    type = DirichletBC
    variable = disp_y
    boundary = left
    value = 0.0
  [../]
  [./fixz1]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value = 0.0
  [../]
  [./fixr1]
    type = DirichletBC
    variable = rot_x
    boundary = left
    value = 0.0
  [../]
  [./fixr2]
    type = DirichletBC
    variable = rot_y
    boundary = left
    value = 0.0
  [../]
  [./fixr3]
    type = DirichletBC
    variable = rot_z
    boundary = left
    value = 0.0
  [../]
[]

[NodalKernels]
  [./force_y2]
    type = UserForcingFunctionNodalKernel
    variable = disp_y
    boundary = right
    function = force
  [../]
[]

[Functions]
  [./force]
    type = PiecewiseLinear
    x = '0.0 0.2 0.4 10.0'
    y = '0.0 0.01  0.0  0.0'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  line_search = 'none'
  l_tol = 1e-11
  nl_max_its = 15
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-10
  start_time = 0.0
  dt = 0.2
  end_time = 5.0
  timestep_tolerance = 1e-6

  # Time integrator
  [./TimeIntegrator]
    type = NewmarkBeta
    beta = 0.4225
    gamma = 0.8
  [../]
[]

[Kernels]
  [./solid_disp_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 0
    variable = disp_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 1
    variable = disp_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 2
    variable = disp_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 3
    variable = rot_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 4
    variable = rot_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 5
    variable = rot_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./inertial_force_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 0
    variable = disp_x
    alpha = -0.3
  [../]
  [./inertial_force_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 1
    variable = disp_y
    alpha = -0.3
  [../]
  [./inertial_force_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 2
    variable = disp_z
    alpha = -0.3
  [../]
  [./inertial_force_rot_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 3
    variable = rot_x
    alpha = -0.3
  [../]
  [./inertial_force_rot_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 4
    variable = rot_y
    alpha = -0.3
  [../]
  [./inertial_force_rot_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 5
    variable = rot_z
    alpha = -0.3
  [../]
[]

[Materials]
  [./elasticity]
    type = ComputeElasticityBeam
    youngs_modulus = 1.0e4
    poissons_ratio = -0.999875
    shear_coefficient = 1.0
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalBeamStrain
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    area = 0.01
    Ay = 0.0
    Az = 0.0
    Iy = 1.0e-4
    Iz = 1.0e-4
    y_orientation = '0.0 1.0 0.0'
  [../]
  [./stress]
    type = ComputeBeamResultants
    block = 0
  [../]
  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1.0'
  [../]
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_x
  [../]
  [./disp_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_y
  [../]
  [./vel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = vel_y
  [../]
  [./accel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = accel_y
  [../]
[]

[Outputs]
  file_base = 'dyn_euler_small_rayleigh_hht_out'
  exodus = true
  csv = true
  perf_graph = true
[]

modules/tensor_mechanics/test/tests/dynamics/wave_1D/wave_rayleigh_hht_ti.i

# Wave propogation in 1D using HHT time integration in the presence of Rayleigh damping
#
# The test is for an 1D bar element of length 4m  fixed on one end
# with a sinusoidal pulse dirichlet boundary condition applied to the other end.
# alpha, beta and gamma are HHT  time integration parameters
# eta and zeta are mass dependent and stiffness dependent Rayleigh damping
# coefficients, respectively.
# The equation of motion in terms of matrices is:
#
# M*accel + (eta*M+zeta*K)*((1+alpha)*vel-alpha*vel_old)
# +(1+alpha)*K*disp-alpha*K*disp_old = 0
#
# Here M is the mass matrix, K is the stiffness matrix
#
# The displacement at the first, second, third and fourth node at t = 0.1 are
# -7.787499960311491942e-02, 1.955566679096475483e-02 and -4.634888180231294501e-03, respectively.

[Mesh]
  type = GeneratedMesh
  dim = 3
  nx = 1
  ny = 4
  nz = 1
  xmin = 0.0
  xmax = 0.1
  ymin = 0.0
  ymax = 4.0
  zmin = 0.0
  zmax = 0.1
[]


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

[AuxVariables]
  [./vel_x]
  [../]
  [./accel_x]
  [../]
  [./vel_y]
  [../]
  [./accel_y]
  [../]
  [./vel_z]
  [../]
  [./accel_z]
  [../]
[]

[Kernels]
  [./DynamicTensorMechanics]
    displacements = 'disp_x disp_y disp_z'
    alpha = -0.3
    zeta = 0.1
  [../]
  [./inertia_x]
    type = InertialForce
    variable = disp_x
    eta=0.1
    alpha = -0.3
  [../]
  [./inertia_y]
    type = InertialForce
    variable = disp_y
    eta=0.1
    alpha = -0.3
  [../]
  [./inertia_z]
    type = InertialForce
    variable = disp_z
    eta = 0.1
    alpha = -0.3
  [../]

[]

[AuxKernels]
  [./accel_x] # These auxkernels are only to check output
    type = TestNewmarkTI
    displacement = disp_x
    variable = accel_x
    first = false
  [../]
  [./accel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = accel_y
    first = false
  [../]
  [./accel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = accel_z
    first = false
  [../]
  [./vel_x]
    type = TestNewmarkTI
    displacement = disp_x
    variable = vel_x
  [../]
  [./vel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = vel_y
  [../]
  [./vel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = vel_z
  [../]
[]


[BCs]
  [./top_y]
    type = DirichletBC
    variable = disp_y
    boundary = top
    value=0.0
  [../]
  [./top_x]
   type = DirichletBC
    variable = disp_x
    boundary = top
    value=0.0
  [../]
  [./top_z]
    type = DirichletBC
    variable = disp_z
    boundary = top
    value=0.0
  [../]
  [./right_x]
    type = DirichletBC
    variable = disp_x
    boundary = right
    value=0.0
  [../]
  [./right_z]
    type = DirichletBC
    variable = disp_z
    boundary = right
    value=0.0
  [../]
  [./left_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value=0.0
  [../]
  [./left_z]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value=0.0
  [../]
  [./front_x]
    type = DirichletBC
    variable = disp_x
    boundary = front
    value=0.0
  [../]
  [./front_z]
    type = DirichletBC
    variable = disp_z
    boundary = front
    value=0.0
  [../]
  [./back_x]
    type = DirichletBC
    variable = disp_x
    boundary = back
    value=0.0
  [../]
  [./back_z]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value=0.0
  [../]
  [./bottom_x]
    type = DirichletBC
    variable = disp_x
    boundary = bottom
    value=0.0
  [../]
  [./bottom_z]
    type = DirichletBC
    variable = disp_z
    boundary = bottom
    value=0.0
  [../]
  [./bottom_y]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = bottom
    function = displacement_bc
  [../]
[]

[Materials]
  [./Elasticity_tensor]
    type = ComputeElasticityTensor
    block = 0
    fill_method = symmetric_isotropic
    C_ijkl = '1 0'
  [../]

  [./strain]
    type = ComputeSmallStrain
    block = 0
    displacements = 'disp_x disp_y disp_z'
  [../]

  [./stress]
    type = ComputeLinearElasticStress
    block = 0
  [../]

  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1'
  [../]

[]

[Executioner]
  type = Transient
  start_time = 0
  end_time = 6.0
  l_tol = 1e-12
  nl_rel_tol = 1e-12
  dt = 0.1

  [./TimeIntegrator]
    type = NewmarkBeta
    beta = 0.422
    gamma = 0.8
  [../]
[]


[Functions]
  [./displacement_bc]
    type = PiecewiseLinear
    data_file = 'sine_wave.csv'
    format = columns
  [../]

[]

[Postprocessors]
  [./_dt]
    type = TimestepSize
  [../]
  [./disp_1]
    type = NodalVariableValue
    nodeid = 1
    variable = disp_y
  [../]
  [./disp_2]
    type = NodalVariableValue
    nodeid = 3
    variable = disp_y
  [../]
  [./disp_3]
    type = NodalVariableValue
    nodeid = 10
    variable = disp_y
  [../]
  [./disp_4]
    type = NodalVariableValue
    nodeid = 14
    variable = disp_y
  [../]
[]

[Outputs]
  file_base = 'wave_rayleigh_hht_out'
  exodus = true
  perf_graph = true
[]

test/tests/kernels/vector_dot_dot/vector_test.i

# Tests calculation of first and second time derivative
# of a coupled vector variable in a material
# a_vec(x,y,z,t) = [t*(t*x + y), t*y, 0]
# a_vec_dot(x,y,z,t) = [2*t*x + y, y, 0]
# a_vec_dot_dot(x,y,z,t) = [2*x, 0, 0]
#
# IMPORTANT NOTE:
# Currently, this test produces a_vec_dot and a_vec_dot_dot that contains oscillations over time.
# This is a known by-product of Newmark Beta time integration (see the Newmark Beta documentation),
# but as of Summer 2019, there is no alternative time integrator in MOOSE that can dampen these
# oscillations. This test is used as coverage for the function call coupledVectorDotDot.

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 4
  ymin = 0
  ymax = 4
  nx = 8
  ny = 8
[]

[Functions]
  [a_fn]
    type = ParsedVectorFunction
    value_x = 't * (t * x + y)'
    value_y = 't * y'
    value_z = 0
  []
[]

[AuxVariables]
  [a]
    family = LAGRANGE_VEC
    order = FIRST
  []
[]

[AuxKernels]
  [a_ak]
    type = VectorFunctionAux
    variable = a
    function = a_fn
  []
[]

[Materials]
  [cm]
    type = VectorCoupledValuesMaterial
    variable = a
  []
[]

[Variables]
  [u]   # u is zero
    family = LAGRANGE_VEC
    order = FIRST
  []
[]

[Kernels]
  [td]
    type = VectorTimeDerivative
    variable = u
  []
[]

[Executioner]
  type = Transient
  dt = 1
  num_steps = 3

  [TimeIntegrator]
    type = NewmarkBeta
  []
[]

[Outputs]
  [./out]
    type = Exodus
    output_material_properties = true
    show_material_properties = 'a_value a_dot a_dot_dot a_dot_du a_dot_dot_du'
  [../]
[]

test/tests/variables/time_derivatives_neighbor/test.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 4
  nx = 2
[]

[Functions]
  [a_fn]
    type = ParsedFunction
    value = 't*(t+x)'
  []
[]

[AuxVariables]
  [a]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[AuxKernels]
  [a_ak]
    type = FunctionAux
    variable = a
    function = a_fn
  []
[]

[Materials]
  [cm]
    type = CoupledValuesMaterial
    variable = a
  []
[]

[Variables]
  [u]
    family = MONOMIAL
    order = CONSTANT
  []
[]

[Kernels]
  [td]
    type = TimeDerivative
    variable = u
  []
[]

[DGKernels]
  [dgk]
    type = MatDGKernel
    variable = u
    mat_prop = a_value
  []
[]

[Executioner]
  type = Transient
  dt = 1
  num_steps = 3

  [TimeIntegrator]
    type = NewmarkBeta
  []
  [Quadrature]
    type = GAUSS
    order = FIRST
  []
[]

[Outputs]
  [./out]
    type = Exodus
    output_material_properties = true
    show_material_properties = 'a_value a_dot a_dot_dot a_dot_du a_dot_dot_du'
  [../]
[]

test/tests/time_integrators/newmark-beta/newmark_beta_prescribed_parameters.i

###########################################################
# This is a simple test with a time-dependent problem
# demonstrating the use of the TimeIntegrator system.
#
# Testing that the first and second time derivatives
# are calculated correctly using the Newmark-Beta method
#
# @Requirement F1.30
###########################################################

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

[Variables]
  [u]
  []
[]

[Functions]
  [forcing_fn]
    type = PiecewiseLinear
    x = '0.0 0.1 0.2    0.3  0.4    0.5  0.6'
    y = '0.0 0.0 0.0025 0.01 0.0175 0.02 0.02'
  []
[]

[Kernels]
  [ie]
    type = TimeDerivative
    variable = u
  []

  [diff]
    type = Diffusion
    variable = u
  []
[]

[BCs]
  [left]
    type = FunctionDirichletBC
    variable = u
    boundary = 'left'
    function = forcing_fn
  []
  [right]
    type = FunctionDirichletBC
    variable = u
    boundary = 'right'
    function = forcing_fn
  []
[]

[Executioner]
  type = Transient
  start_time = 0.0
  num_steps = 6
  dt = 0.1
  [TimeIntegrator]
    type = NewmarkBeta
    beta = 0.4225
    gamma = 0.8
  []
[]

[Postprocessors]
  [udot]
    type = ElementAverageTimeDerivative
    variable = u
  []
  [udotdot]
    type = ElementAverageSecondTimeDerivative
    variable = u
  []
  [u]
    type = ElementAverageValue
    variable = u
  []
[]

[Outputs]
  csv = true
[]

References

K. J. Bathe and G. Noh. Insight into an implicit time integration scheme for structural dynamics. Computers and Structures, 98-99:1–6, 2012.

@article{bathe2012insight,
    author = "Bathe, K. J. and Noh, G.",
    title = "Insight into an implicit time integration scheme for structural dynamics",
    journal = "Computers and Structures",
    volume = "98-99",
    pages = "1-6",
    year = "2012"
}

N. M. Newmark. A method of computation for structural dynamics. Journal of Engineering Mechanics, 85(EM3):67–94, 1959.

@article{newmark1959amethod,
    author = "Newmark, N. M.",
    title = "A method of computation for structural dynamics",
    journal = "Journal of Engineering Mechanics",
    publisher = "ASCE",
    volume = "85",
    number = "EM3",
    pages = "67--94",
    year = "1959"
}

Description
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