FFTMechanics

deGeus variational mechanics solve. Updates the coupled buffer holding the deformation gradient tensor.

Solve a heterogeneous mechanics problem using the approach by deGeus et al. de Geus et al. (2017).

Example Input File Syntax

[TensorComputes]
  [Solve]
    [root]
      [applied_strain]
        type = MacroscopicShearTensor
        buffer = applied_strain
        F = F
      []
      [hyper_elasticity]
        type = HyperElasticIsotropic
        buffer = stress
        F = Fnew
        K = K
        mu = mu
      []
      [mech]
        type = FFTMechanics
        buffer = Fnew
        F = F
        K = K
        mu = mu
        l_tol = 1e-2
        nl_rel_tol = 2e-2
        nl_abs_tol = 2e-2
        constitutive_model = hyper_elasticity
        stress = stress
        tangent_operator = dstressdstrain
        applied_macroscopic_strain = applied_strain
      []
    []
  []
[]
!listing-end

Notes: - Supplies and updates the deformation gradient buffer Fnew. - Requires a constitutive-model compute that provides stress and its tangent. - An optional macroscopic strain buffer lets you impose a prescribed average deformation.

Input Parameters

KBulk modulus
C++ Type:std::string
Controllable:No
Description:Bulk modulus
bufferThe buffer this compute is writing to
C++ Type:std::string
Controllable:No
Description:The buffer this compute is writing to
constitutive_modelTensor compute for the constitutive model (computes stress from displacement gradeint tensor)
C++ Type:std::string
Controllable:No
Description:Tensor compute for the constitutive model (computes stress from displacement gradeint tensor)

Required Parameters

FFDeformation gradient tensor.
Default:F
C++ Type:std::string
Controllable:No
Description:Deformation gradient tensor.
applied_macroscopic_strainApplied macroscopic strain
C++ Type:std::string
Controllable:No
Description:Applied macroscopic strain
l_max_itsMaximum number of congugate gradient solve iterations
C++ Type:unsigned int
Controllable:No
Description:Maximum number of congugate gradient solve iterations
l_tol0.01Linear congugate gradient solve tolerance
Default:0.01
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Linear congugate gradient solve tolerance
muShear modulus
C++ Type:std::string
Controllable:No
Description:Shear modulus
nl_abs_tol1e-08Nonlinear solve relative tolerance
Default:1e-08
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Nonlinear solve relative tolerance
nl_max_its100Maximum number of nonlinear solve iterations
Default:100
C++ Type:unsigned int
Controllable:No
Description:Maximum number of nonlinear solve iterations
nl_rel_tol1e-05Nonlinear solve absolute tolerance
Default:1e-05
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Nonlinear solve absolute tolerance
stressstressComputed stress
Default:stress
C++ Type:std::string
Controllable:No
Description:Computed stress
tangent_operatordstressdstrainTangent operator
Default:dstressdstrain
C++ Type:std::string
Controllable:No
Description:Tangent operator
verboseFalsePrint non-linear residuals.
Default:False
C++ Type:bool
Controllable:No
Description:Print non-linear residuals.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

References

T.W.J. de Geus, J. Vondřejc, J. Zeman, R.H.J. Peerlings, and M.G.D. Geers. Finite strain fft-based non-linear solvers made simple. Computer Methods in Applied Mechanics and Engineering, 318:412–430, 2017. URL: https://www.sciencedirect.com/science/article/pii/S0045782516318709, doi:https://doi.org/10.1016/j.cma.2016.12.032.

@article{DEGEUS2017412,
    author = "{de Geus}, T.W.J. and Vondřejc, J. and Zeman, J. and Peerlings, R.H.J. and Geers, M.G.D.",
    title = "Finite strain FFT-based non-linear solvers made simple",
    journal = "Computer Methods in Applied Mechanics and Engineering",
    volume = "318",
    pages = "412-430",
    year = "2017",
    issn = "0045-7825",
    doi = "https://doi.org/10.1016/j.cma.2016.12.032",
    url = "https://www.sciencedirect.com/science/article/pii/S0045782516318709",
    keywords = "Homogenization, Micromechanics, Fast Fourier Transform (FFT), Representative Volume Element (RVE), Finite strains",
    abstract = "Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model."
}

(examples/degeus_mechanics/mech.i)

[Domain]
  dim = 3
  nx = 32
  ny = 32
  nz = 32
  xmax = ${fparse 2*pi}
  ymax = ${fparse 2*pi}
  zmax = ${fparse 2*pi}
  mesh_mode = DUMMY
[]

[TensorComputes]
  [Initialize]
    [Finit]
      type = RankTwoIdentity
      buffer = F
    []

    [phase]
      type = PhaseMechanicsTest
      buffer = phase
    []
    [K]
      type = ParsedCompute
      buffer = K
      expression = '(1-phase)*Ka + phase*Kb'
      inputs = phase
      constant_names = 'Ka Kb'
      constant_expressions = '0.833 8.33'
    []
    [mu]
      type = ParsedCompute
      buffer = mu
      expression = '(1-phase)*mua + phase*mub'
      inputs = phase
      constant_names = 'mua mub'
      constant_expressions = '0.386 3.86'
    []
  []

  [Solve]
    [hyper_elasticity]
      type = HyperElasticIsotropic
      buffer = stress
      F = Fnew
      K = K
      mu = mu
    []

    [root]
      [applied_strain]
        type = MacroscopicShearTensor
        buffer = applied_strain
      []
      [mech]
        type = FFTMechanics
        buffer = Fnew
        F = F
        K = K
        mu = mu
        l_tol = 1e-2
        nl_rel_tol = 2e-2
        nl_abs_tol = 2e-2
        constitutive_model = hyper_elasticity
        stress = stress
        applied_macroscopic_strain = applied_strain
      []
    []
  []

  [Postprocess]
    [displacements]
      type = ComputeDisplacements
      buffer = disp
      F = F
    []
    [vonmises]
      type = ComputeVonMisesStress
      buffer = sV
    []
  []
[]

[TensorSolver]
  # no variables are integrated by this solver (FFTMechanics performs a steady state mechanics solve)
  type = ForwardEulerSolver
  root_compute = root
  # deformation tensor is just forwarded Fnew -> F
  forward_buffer = F
  forward_buffer_new = Fnew
  substeps = 10
[]

[TensorOutputs]
  [deformation_tensor]
    type = XDMFTensorOutput
    buffer = 'disp sV F'
    output_mode = 'OVERSIZED_NODAL CELL CELL'
    enable_hdf5 = true
  []
[]

[Executioner]
  type = Transient
  num_steps = 100
  dt = 0.01
[]

(test/tests/mechanics/mech.i)

[Domain]
  dim = 2
  nx = 32
  ny = 32
  xmax = ${fparse 2*pi}
  ymax = ${fparse 2*pi}
  zmax = ${fparse 2*pi}
  mesh_mode = DUMMY
  parallel_mode = FFT_AUTO
[]

[TensorComputes]
  [Initialize]
    [phase]
      type = ParsedCompute
      expression = '(cos(x)/2+0.5)^1*(cos(y)/2+0.5)^1*(cos(z)/2+0.5)^1'
      extra_symbols = true
      buffer = phase
    []
    [K]
      type = ParsedCompute
      buffer = K
      expression = '(1-phase)*Ka + phase*Kb'
      inputs = phase
      constant_names = 'Ka Kb'
      constant_expressions = '1 10'
    []
    [mu]
      type = ParsedCompute
      buffer = mu
      expression = '(1-phase)*mua + phase*mub'
      inputs = phase
      constant_names = 'mua mub'
      constant_expressions = '0.5 5'
    []
    [Finit]
      type = RankTwoIdentity
      buffer = F
    []
  []

  [Solve]
    [hyper_elasticity]
      type = HyperElasticIsotropic
      buffer = stress
      F = Fnew
      K = K
      mu = mu
    []

    [root]
      [applied_strain]
        type = MacroscopicShearTensor
        buffer = applied_strain
      []
      [mech]
        type = FFTMechanics
        buffer = Fnew
        F = F
        K = K
        mu = mu
        l_max_its = 40
        l_tol = 1e-5
        nl_rel_tol = 2e-4
        nl_abs_tol = 2e-3
        constitutive_model = hyper_elasticity
        stress = stress
        applied_macroscopic_strain = applied_strain
      []
    []
  []

  [Postprocess]
    [displacements]
      type = ComputeDisplacements
      buffer = disp
      F = F
    []
    [vonmises]
      type = ComputeVonMisesStress
      buffer = sV
    []
  []
[]

[TensorSolver]
  # no variables are integrated by this solver (FFTMechanics performs a steady state mechanics solve)
  type = ForwardEulerSolver
  root_compute = root
  # deformation tensor is just forwarded Fnew -> F
  forward_buffer = F
  forward_buffer_new = Fnew
  substeps = 3
[]

[TensorOutputs]
  active = 'parallel' # serial

  [serial]
    type = XDMFTensorOutput
    file_base = mech_serial
    buffer = 'disp sV F phase'
    output_mode = 'OVERSIZED_NODAL CELL CELL NODE'
    enable_hdf5 = true
    execute_on = 'TIMESTEP_END'
  []
  [parallel]
    # NODE and OVERSIZED_NODAL are not yet available in parallel output
    type = XDMFTensorOutput
    file_base = mech_parallel
    buffer = 'sV F phase'
    output_mode = 'CELL CELL CELL'
    enable_hdf5 = true
    execute_on = 'TIMESTEP_END'
  []
[]


[Executioner]
  type = Transient
  num_steps = 3
  dt = 0.02
[]

(test/tests/mechanics/mech3d.i)

[Domain]
  dim = 3
  nx = 16
  ny = 16
  nz = 16
  xmax = ${fparse 2*pi}
  ymax = ${fparse 2*pi}
  zmax = ${fparse 2*pi}
  mesh_mode = DUMMY
  parallel_mode = FFT_AUTO
[]

[TensorComputes]
  [Initialize]
    [phase]
      type = ParsedCompute
      expression = '(cos(x)/2+0.5)^1*(cos(y)/2+0.5)^1*(cos(z)/2+0.5)^1'
      extra_symbols = true
      buffer = phase
    []
    [K]
      type = ParsedCompute
      buffer = K
      expression = '(1-phase)*Ka + phase*Kb'
      inputs = phase
      constant_names = 'Ka Kb'
      constant_expressions = '1 10'
    []
    [mu]
      type = ParsedCompute
      buffer = mu
      expression = '(1-phase)*mua + phase*mub'
      inputs = phase
      constant_names = 'mua mub'
      constant_expressions = '0.5 5'
    []
    [Finit]
      type = RankTwoIdentity
      buffer = F
    []
  []

  [Solve]
    [hyper_elasticity]
      type = HyperElasticIsotropic
      buffer = stress
      F = Fnew
      K = K
      mu = mu
    []

    [root]
      [applied_strain]
        type = MacroscopicShearTensor
        buffer = applied_strain
      []
      [mech]
        type = FFTMechanics
        buffer = Fnew
        F = F
        K = K
        mu = mu
        l_tol = 1e-2
        nl_rel_tol = 2e-2
        nl_abs_tol = 2e-2
        constitutive_model = hyper_elasticity
        stress = stress
        applied_macroscopic_strain = applied_strain
      []
    []
  []

  [Postprocess]
    [displacements]
      type = ComputeDisplacements
      buffer = disp
      F = F
    []
    [vonmises]
      type = ComputeVonMisesStress
      buffer = sV
    []
  []
[]

[TensorSolver]
  # no variables are integrated by this solver (FFTMechanics performs a steady state mechanics solve)
  type = ForwardEulerSolver
  root_compute = root
  # deformation tensor is just forwarded Fnew -> F
  forward_buffer = F
  forward_buffer_new = Fnew
  substeps = 10
[]

[TensorOutputs]
  active = 'parallel' # serial

  [serial]
    type = XDMFTensorOutput
    file_base = mech3d_serial
    buffer = 'disp sV F phase'
    output_mode = 'OVERSIZED_NODAL CELL CELL NODE'
    enable_hdf5 = true
    execute_on = 'TIMESTEP_END'
  []
  [parallel]
    # NODE and OVERSIZED_NODAL are not yet available in parallel output
    type = XDMFTensorOutput
    file_base = mech3d_parallel
    buffer = 'sV F phase'
    output_mode = 'CELL CELL CELL'
    enable_hdf5 = true
    execute_on = 'TIMESTEP_END'
  []
[]

[Executioner]
  type = Transient
  num_steps = 3
  dt = 0.01
[]

Example Input File Syntax
Input Parameters
Input Files
References