getfem_plasticity.h

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/* -*- c++ -*- (enables emacs c++ mode) */
/*===========================================================================
 
 Copyright (C) 2002-2020 Amandine Cottaz, Yves Renard
 Copyright (C) 2014-2020 Konstantinos Poulios
 
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/**@file getfem_plasticity.h
   @author  Yves Renard <[email protected]>,
   @author  Julien Pommier <[email protected]>
   @author  Amandine Cottaz
   @date June 2, 2010.
   @brief Plasticty bricks.
*/
#ifndef GETFEM_PLASTICITY_H__
#define GETFEM_PLASTICITY_H__
 
#include "getfem_models.h"
#include "getfem_assembling_tensors.h"
#include "getfem_derivatives.h"
#include "getfem_interpolation.h"
#include "gmm/gmm_dense_qr.h"
 
namespace getfem {
 
  enum plasticity_unknowns_type {
    DISPLACEMENT_ONLY = 0,
    DISPLACEMENT_AND_PLASTIC_MULTIPLIER = 1,
    DISPLACEMENT_AND_PRESSURE = 2,
    DISPLACEMENT_AND_PLASTIC_MULTIPLIER_AND_PRESSURE = 3
  };
 
  //=================================================================
  //  Small strain Elastoplasticity Brick
  //=================================================================
 
  /**
     Adds a small strain plasticity term to the model `md`. This is the
     main GetFEM brick for small strain plasticity. `lawname` is the name
     of an implemented plastic law, `unknowns_type` indicates the choice
     between a discretization where the plastic multiplier is an unknown of
     the problem or (return mapping approach) just a data of the model
     stored for the next iteration. Remember that in both cases, a multiplier
     is stored anyway. `varnames` is a set of variable and data names with
     length which may depend on the plastic law (at least the displacement,
     the plastic multiplier and the plastic strain). `params` is a list of
     expressions for the parameters (at least elastic coefficients and the
     yield stress). These expressions can be some data names (or even
     variable names) of the model but can also be any scalar valid expression
     of the high level assembly language (such as "1/2", "2+sin(X[0])",
     "1+Norm(v)" ...). The last two parameters optionally provided in
     `params` are the `theta` parameter of the `theta`-scheme (generalized 
     trapezoidal rule) used for the plastic strain integration and the
     time-step`dt`. The default value for `theta` if omitted is 1, which
     corresponds to the classical Backward Euler scheme which is first order
     consistent. `theta=1/2` corresponds to the Crank-Nicolson scheme
     (trapezoidal rule) which is second order consistent. Any value
     between 1/2 and 1 should be a valid value. The default value of `dt` is
     'timestep' which simply indicates the time step defined in the model
     (by md.set_time_step(dt)). Alternatively it can be any expression
     (data name, constant value ...). The time step can be altered from one
     iteration to the next one. `region` is a mesh region.
 
     The available plasticity laws are:
 
     - "Prandtl Reuss" (or "isotropic perfect plasticity").
       Isotropic elasto-plasticity with no hardening. The variables are the
       displacement, the plastic multiplier and the plastic strain.
       The displacement should be a variable and have a corresponding data
       having the same name preceded by "Previous_" corresponding to the
       displacement at the previous time step (typically "u" and "Previous_u").
       The plastic multiplier should also have two versions (typically "xi"
       and "Previous_xi") the first one being defined as data if
       `unknowns_type = DISPLACEMENT_ONLY` or as a variable if
       `unknowns_type = DISPLACEMENT_AND_PLASTIC_MULTIPLIER`.
       The plastic strain should represent a n x n data tensor field stored
       on mesh_fem or (preferably) on an im_data (corresponding to `mim`).
       The data are the first Lame coefficient, the second one (shear modulus)
       and the uniaxial yield stress. IMPORTANT: Note that this law implements
       the 3D expressions. If it is used in 2D, the expressions are just
       transposed to the 2D. For the plane strain approximation, see below.
     - "plane strain Prandtl Reuss"
       (or "plane strain isotropic perfect plasticity")
       The same law as the previous one but adapted to the plane strain
       approximation. Can only be used in 2D.
     - "Prandtl Reuss linear hardening"
       (or "isotropic plasticity linear hardening").
       Isotropic elasto-plasticity with linear isotropic and kinematic
       hardening. An additional variable compared to "Prandtl Reuss" law:
       the accumulated plastic strain. Similarly to the plastic strain, it
       is only stored at the end of the time step, so a simple data is
       required (preferably on an im_data).
       Two additional parameters: the kinematic hardening modulus and the
       isotropic one. 3D expressions only.
     - "plane strain Prandtl Reuss linear hardening"
       (or "plane strain isotropic plasticity linear hardening").
       The same law as the previous one but adapted to the plane strain
       approximation. Can only be used in 2D.
 
     See GetFEM user documentation for further explanations on the
     discretization of the plastic flow and on the implemented plastic laws.
     See also GetFEM user documentation on time integration strategy
     (integration of transient problems).
 
     IMPORTANT : remember that `small_strain_elastoplasticity_next_iter` has
     to be called at the end of each time step, before the next one
     (and before any post-treatment : this sets the value of the plastic
     strain and plastic multiplier).
   */
  size_type add_small_strain_elastoplasticity_brick
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   size_type region = size_type(-1));
  
  /** Function that allows to pass from a time step to another for the
      small strain plastic brick. The parameters have to be exactly the
      same as the ones of the `add_small_strain_elastoplasticity_brick`,
      so see the documentation of this function for any explanations.
      Basically, this brick computes the plastic strain and the plastic
      multiplier and stores them for the next step. Additionaly, it copies
      the computed displacement to the data that stores the displacement
      of the previous time step (typically "u" to "Previous_u").
      It has to be called before any use of
      `compute_small_strain_elastoplasticity_Von_Mises`.
   */
  void small_strain_elastoplasticity_next_iter
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   size_type region = size_type(-1)) ;
 
  /** This function computes the Von Mises stress field with respect to
      a small strain elastoplasticity term, approximated on `mf_vm`,
      and stores the result into `VM`.  All other parameters have to be
      exactly the same as for `add_small_strain_elastoplasticity_brick`.
      Remember that `small_strain_elastoplasticity_next_iter` has to be called
      before any call of this function.
   */
  void compute_small_strain_elastoplasticity_Von_Mises
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   const mesh_fem &mf_vm, model_real_plain_vector &VM,
   size_type region = size_type(-1));
 
 
  //=================================================================
  // Abstract contraints projection
  //=================================================================
 
 
  /** Abstract projection of a stress tensor onto a set of admissible
      stress tensors.
  */
  class abstract_constraints_projection {
  protected :
    size_type flag_hyp;
 
  public :
      /* if flag_proj=0 the output will be Proj(tau)
       * if flag_proj=1 the output will be gradProj(tau)
       * no others values allowed for flag_proj
       */
    virtual void do_projection(const base_matrix& tau,
                               scalar_type stress_threshold,
                               base_matrix& proj,
                               size_type flag_proj) const = 0;
    abstract_constraints_projection (size_type flag_hyp_ = 0) :
      flag_hyp(flag_hyp_) {}
    virtual ~abstract_constraints_projection () {}
  };
 
  typedef std::shared_ptr<const abstract_constraints_projection>
  pconstraints_projection;
 
  //=================================================================
  // Von Mises projection
  //=================================================================
 
 
  /** Von Mises projection */
  class VM_projection : public abstract_constraints_projection {
 
    /* used to compute the projection */
    template<typename MAT>
    void tau_m_Id(const MAT& tau, MAT &taumid) const {
      scalar_type trace = gmm::mat_trace(tau);
      size_type size_of_tau = gmm::mat_nrows(tau);
      gmm::copy(gmm::identity_matrix(),taumid);
      gmm::scale(taumid, trace / scalar_type(size_of_tau));
    }
 
    /* used to compute the projection */
    template<typename MAT>
    void tau_d(const MAT& tau, MAT &taud) const {
      tau_m_Id(tau, taud);
      gmm::scale(taud, scalar_type(-1));
      gmm::add(tau, taud);
    }
 
 
    public :
 
    /** the Von Mises projection computation */
    /* on input : tau matrix, on output : the projection of tau */
    virtual void do_projection(const base_matrix& tau,
                               scalar_type stress_threshold,
                               base_matrix& proj,
                               size_type flag_proj)  const {
 
      /* be sure that flag_proj has a correct value */
      GMM_ASSERT1(flag_proj == 0 || flag_proj ==1,
                  "wrong value for the projection flag, "
                  "must be 0 or 1 ");
 
      /* be sure that stress_threshold has a correct value */
      GMM_ASSERT1(stress_threshold>=0., "s is not a positive number "
                  << stress_threshold << ". You need to set "
                  << "s as a positive number");
 
      size_type N = gmm::mat_nrows(tau);
      size_type projsize = (flag_proj == 0) ? N : gmm::sqr(N);
      scalar_type normtaud;
 
      /* calculate tau_m*Id */
      base_matrix taumId(N, N);
      tau_m_Id(tau, taumId);
 
      // calcul du deviateur de tau, taud
      base_matrix taud(N,N);
      gmm::add(gmm::scaled(taumId, scalar_type(-1)), tau, taud);
 
      /* plane constraints */
      if (flag_hyp == 1) {  // To be done ...
        N /= 2;
        GMM_ASSERT1(!N, "wrong value for CALCULATION HYPOTHESIS, "
                    "must be /=1 SINCE n/=2");
        // we form the 3D tau tensor considering
        // that tau(3,j)=tau(i,3)=0
        base_matrix tau_aux(3,3); gmm::clear(tau_aux);
        gmm::copy(tau,gmm::sub_matrix
                  (tau_aux,gmm::sub_interval(0,2)));
        // we calculate tau deviator and its norms
        base_matrix taud_aux(3,3);
        tau_d(tau_aux, taud_aux);
        normtaud=gmm::mat_euclidean_norm(taud_aux);
      }
      else normtaud=gmm::mat_euclidean_norm(taud);
 
 
      /* dimension and initialization of proj matrix or
         its derivative */
      gmm::resize(proj, projsize, projsize);
 
      if (normtaud <= stress_threshold) {
        switch(flag_proj) {
        case 0: gmm::copy(tau, proj); break;
        case 1: gmm::copy(gmm::identity_matrix(), proj); break;
        }
      }
      else {
        switch(flag_proj) {
        case 0:
          gmm::copy(gmm::scaled(taud, stress_threshold/normtaud),
                    proj);
          gmm::add(taumId,proj);
          break;
        case 1:
          base_matrix Igrad(projsize, projsize);
          gmm::copy(gmm::identity_matrix(),Igrad);
          base_matrix Igrad2(projsize, projsize);
 
          // build vector[1 0 0 1  0 0 1...] to be copied in certain
          // columns of Igrad(*)Igrad
          base_vector aux(projsize);
          for (size_type i=0; i < N; ++i)
            aux[i*N + i] = scalar_type(1);
 
          // Copy in a selection of columns of Igrad(*)Igrad
          for (size_type i=0; i < N; ++i)
            gmm::copy(aux, gmm::mat_col(Igrad2, i*N + i));
 
          // Compute Id_grad
          base_matrix Id_grad(projsize, projsize);
          scalar_type rr = scalar_type(1)/scalar_type(N);
          gmm::copy(gmm::scaled(Igrad2, -rr), Id_grad);
          gmm::add(Igrad, Id_grad);
 
 
          // Compute ngrad(*)ngrad
          base_matrix ngrad2(projsize, projsize);
          // Compute the normal n
          base_matrix un(N, N);
          gmm::copy(gmm::scaled(taud, 1./normtaud),un);
 
          // Copy of the normal in a column vector
          // in the Fortran order
          std::copy(un.begin(), un.end(), aux.begin());
 
          // Loop on the columns of ngrad(*)ngrad
          for (size_type j=0; j < projsize; ++j)
            gmm::copy(gmm::scaled(aux,aux[j]),
                      gmm::mat_col(ngrad2,j));
 
 
          // Final computation of the projection gradient
          gmm::copy(gmm::identity_matrix(), proj);
          gmm::add(gmm::scaled(ngrad2, scalar_type(-1)), proj);
          base_matrix aux2(projsize, projsize);
          gmm::copy(gmm::scaled(proj, stress_threshold/normtaud),
                    aux2);
          gmm::mult(aux2,Id_grad,proj);
          gmm::add(gmm::scaled(Igrad2, rr),proj);
          break;
        }
      }
    }
 
 
    VM_projection(size_type flag_hyp_ = 0) :
      abstract_constraints_projection (flag_hyp_) {}
  };
 
 
  // Finite strain elastoplasticity
 
  /** Add a linear function with the name specified by `name` to represent
      linear isotropoc hardening in plasticity with initial yield limit
      `sigma_y0` and hardening modulus `H`.
      A true value of the `frobenius` argument will express the hardening
      function in terms of Frobenius norms both for the strain input and
      the stress output, instead of the corresponding Von-Mises quantities.
  */
  void ga_define_linear_hardening_function
  (const std::string &name, scalar_type sigma_y0, scalar_type H, bool frobenius=true);
 
  /** Add a Ramberg-Osgood hardening function with the name specified by
     `name`, for reference stress and strain given by `sigma_ref` and
      `eps_ref` respectively and for a hardening exponent `n`.
      A true value of the `frobenius` argument will express the hardening
      function in terms of Frobenius norms both for the strain input and
      the stress output, instead of the corresponding Von-Mises quantities.
  */
  void ga_define_Ramberg_Osgood_hardening_function
  (const std::string &name,
   scalar_type sigma_ref, scalar_type eps_ref, scalar_type n,
   bool frobenius=false);
 
  /** Add a Ramberg-Osgood hardening function with the name specified by
     `name`, for reference stress `sigma_ref`, Young's modulus `E`,
      offset parameter `alpha` and hardening parameter `n`.
      A true value of the `frobenius` argument will express the hardening
      function in terms of Frobenius norms both for the strain input and
      the stress output, instead of the corresponding Von-Mises quantities.
  */
  inline void ga_define_Ramberg_Osgood_hardening_function
  (const std::string &name, scalar_type sigma_ref, scalar_type E, 
   scalar_type alpha, scalar_type n, bool frobenius=false) {
    ga_define_Ramberg_Osgood_hardening_function
      (name, sigma_ref, alpha*sigma_ref/E, n, frobenius);
  }
 
  /** Add a finite strain elastoplasticity brick to the model.
      For the moment there is only one supported law defined through 
      `lawname` as "Simo_Miehe".
      This law supports to possibilities of unknown variables to solve for
      defined by means of `unknowns_type` set to either
      `DISPLACEMENT_AND_PLASTIC_MULTIPLIER` or
      `DISPLACEMENT_AND_PLASTIC_MULTIPLIER_AND_PRESSURE`
      The  "Simo_Miehe" law expects as `varnames` a vector of
      the following names that have to be defined as variables in the
      model:
      - the displacement variable which has to be defined as an unknown,
      - the plastic multiplier which has also defined as an unknown,
      - optionally the pressure variable for a mixed displacement-pressure
        formulation for `DISPLACEMENT_AND_PLASTIC_MULTIPLIER_AND_PRESSURE`
        as `unknowns_type`,
      - the name of a (scalar) fem_data or im_data field that holds the
        plastic strain at the previous time step, and
      - the name of a fem_data or im_data field that holds all
        non-repeated components of the inverse of the plastic right
        Cauchy-Green tensor at the previous time step
        (it has to be a 4 element vector for plane strain 2D problems
        and a 6 element vector for 3D problems).
      The  "Simo_Miehe" law also expects as `params` a vector of
      the following three parameters:
      - an expression for the initial bulk modulus K,
      - an expression for the initial shear modulus G,
      - the name of a user predefined function that decribes
        the yield limit as a function of the hardening variable
        (both the yield limit and the hardening variable values are
         assumed to be Frobenius norms of appropriate stress and strain
         tensors, respectively),
  */
  size_type add_finite_strain_elastoplasticity_brick
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   size_type region = size_type(-1));
 
  /** This function permits to update the state variables for a finite
      strain elastoplasticity brick, based on the current displacements
      and plastic multiplier fields (optionally also the the pressure field
      in the case of a mixed displacement-pressure formulation).
      The parameters have to be exactly the same as the ones of the
     `add_finite_strain_elastoplasticity_brick`, so see the documentation
      of this function for any explanations.
      Basically, `varnames` contains both the names of the input fields as
      well as the names of the fields to be updated.
  */
  void finite_strain_elastoplasticity_next_iter
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   size_type region = size_type(-1));
 
  /** This function computes the Von Mises stress field with respect to
      a finite strain elastoplasticity term, approximated on `mf_vm`,
      and stores the result into `VM`.  All other parameters have to be
      exactly the same as for `add_finite_strain_elastoplasticity_brick`.
   */
  void compute_finite_strain_elastoplasticity_Von_Mises
  (model &md, const mesh_im &mim,
   std::string lawname, plasticity_unknowns_type unknowns_type,
   const std::vector<std::string> &varnames,
   const std::vector<std::string> &params,
   const mesh_fem &mf_vm, model_real_plain_vector &VM,
   size_type region = size_type(-1));
 
 
 
  //=================================================================
  //
  //  Old version of an elastoplasticity Brick for isotropic perfect
  //  plasticity with the low level generic assembly.
  //  Particularity of this brick: the flow rule is integrated on
  //  finite element nodes (not on Gauss points).
  //
  //=================================================================
 
 
  /** Add a nonlinear elastoplasticity term to the model for small
      deformations and an isotropic material, with respect
      to the variable `varname`.
      Note that the constitutive lawtype of projection
      to be used is described by `ACP` which should not be
      freed while the model is used.
      `datalambda` and `datamu` describe the Lamé coeffcients
      of the studied material. Could be scalar or vector fields
      described on a finite element method.
      `datathreshold` represents the elasticity threshold
      of the material. It could be a scalar or a vector field
      described on the same finite element method as
      the Lamé coefficients.
      `datasigma` represents the stress constraints values
      supported by the material. It should be a vector field
      described on a finite element method.
      `previous_dep_name` represents the displacement at the previous time step.
      Moreover, if `varname` is described
      onto a K-th order mesh_fem, `datasigma` has to be described
      on a mesh_fem of order at least K-1.
  */
  size_type add_elastoplasticity_brick(model &md,
                                       const mesh_im &mim,
                                       const pconstraints_projection &ACP,
                                       const std::string &varname,
                                       const std::string &previous_dep_name,
                                       const std::string &datalambda,
                                       const std::string &datamu,
                                       const std::string &datathreshold,
                                       const std::string &datasigma,
                                       size_type region = size_type(-1));
 
  /** This function permits to compute the new stress constraints
      values supported by the material after a load or an unload.
      `varname` is the main unknown of the problem
      (the displacement),
      `previous_dep_name` represents the displacement at the previous time step,
      `ACP` is the type of projection to be used that could only be
      `Von Mises` for the moment,
      `datalambda` and `datamu` are the Lamé coefficients
      of the material,
      `datathreshold` is the elasticity threshold of the material,
      `datasigma` is the vector which will contains the new
      computed values. */
  void elastoplasticity_next_iter(model &md,
                                  const mesh_im &mim,
                                  const std::string &varname,
                                  const std::string &previous_dep_name,
                                  const pconstraints_projection &ACP,
                                  const std::string &datalambda,
                                  const std::string &datamu,
                                  const std::string &datathreshold,
                                  const std::string &datasigma);
 
  /** This function computes on mf_vm the Von Mises or Tresca stress
      of a field for elastoplasticity and return it into the vector VM.
      Note that `datasigma` should be the vector containing the new
      stress constraints values, i.e. after a load or an unload
      of the material. If `tresca` = 'true', the Tresca stress will
      be computed, otherwise it will be the Von Mises one.*/
  void compute_elastoplasticity_Von_Mises_or_Tresca
  (model &md,
   const std::string &datasigma,
   const mesh_fem &mf_vm,
   model_real_plain_vector &VM,
   bool tresca);
 
  /** This function computes on mf_pl the plastic part, that could appear
      after a load and an unload, into the vector `plast`.
      Note that `datasigma` should be the vector containing the new
      stress constraints values, i.e. after a load or an unload
      of the material. */
  void compute_plastic_part(model &md,
                            const mesh_im &mim,
                            const mesh_fem &mf_pl,
                            const std::string &varname,
                            const std::string &previous_dep_name,
                            const pconstraints_projection &ACP,
                            const std::string &datalambda,
                            const std::string &datamu,
                            const std::string &datathreshold,
                            const std::string &datasigma,
                            model_real_plain_vector &plast);
 
 
 
} /* namespace getfem */
 
#endif