getfem_nonlinear_elasticity.cc

/*===========================================================================
 
 Copyright (C) 2000-2020 Yves Renard
 
 This file is a part of GetFEM
 
 GetFEM  is  free software;  you  can  redistribute  it  and/or modify it
 under  the  terms  of the  GNU  Lesser General Public License as published
 by  the  Free Software Foundation;  either version 3 of the License,  or
 (at your option) any later version along with the GCC Runtime Library
 Exception either version 3.1 or (at your option) any later version.
 This program  is  distributed  in  the  hope  that it will be useful,  but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or  FITNESS  FOR  A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License and GCC Runtime Library Exception for more details.
 You  should  have received a copy of the GNU Lesser General Public License
 along  with  this program;  if not, write to the Free Software Foundation,
 Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA.
 
===========================================================================*/
 
 
#include "getfem/getfem_models.h"
#include "getfem/getfem_nonlinear_elasticity.h"
#include "getfem/getfem_generic_assembly.h"
 
namespace getfem {
 
 
  /* Useful functions to compute the invariants and their derivatives
     Note that the second derivative is symmetrized (see the user
     documentation for more details). The matrix E is assumed to be symmetric.
  */
 
 
  static scalar_type frobenius_product_trans(const base_matrix &A,
                                             const base_matrix &B) {
    size_type N = gmm::mat_nrows(A);
    scalar_type res = scalar_type(0);
    for (size_type i = 0; i < N; ++i)
      for (size_type j = 0; j < N; ++j)
        res += A(i, j) * B(j, i);
    return res;
  }
 
  struct compute_invariants {
 
    const base_matrix &E;
    base_matrix Einv;
    size_type N;
    scalar_type i1_, i2_, i3_, j1_, j2_;
    bool i1_c, i2_c, i3_c, j1_c, j2_c;
 
    base_matrix di1, di2, di3, dj1, dj2;
    bool di1_c, di2_c, di3_c, dj1_c, dj2_c;
 
    base_tensor ddi1, ddi2, ddi3, ddj1, ddj2;
    bool ddi1_c, ddi2_c, ddi3_c, ddj1_c, ddj2_c;
 
 
    /* First invariant tr(E) */
    void compute_i1() {
      i1_ = gmm::mat_trace(E);
      i1_c = true;
    }
 
    void compute_di1() {
      gmm::resize(di1, N, N);
      gmm::copy(gmm::identity_matrix(), di1);
      di1_c = true;
    }
 
    void compute_ddi1() { // not very useful, null tensor
      ddi1 = base_tensor(N, N, N, N);
      ddi1_c = true;
    }
 
    inline scalar_type i1()
    { if (!i1_c) compute_i1(); return i1_; }
 
    inline const base_matrix &grad_i1()
    { if (!di1_c) compute_di1(); return di1; }
 
    inline const base_tensor &sym_grad_grad_i1()
    { if (!ddi1_c) compute_ddi1(); return ddi1; }
 
 
    /* Second invariant (tr(E)^2 - tr(E^2))/2 */
    void compute_i2() {
      i2_ = (gmm::sqr(gmm::mat_trace(E))
             - frobenius_product_trans(E, E)) / scalar_type(2);
      i2_c = true;
    }
 
    void compute_di2() {
      gmm::resize(di2, N, N);
      gmm::copy(gmm::identity_matrix(), di2);
      gmm::scale(di2, i1());
      // gmm::add(gmm::scale(gmm::transposed(E), -scalar_type(1)), di2);
      gmm::add(gmm::scaled(E, -scalar_type(1)), di2);
      di2_c = true;
    }
 
    void compute_ddi2() {
      ddi2 = base_tensor(N, N, N, N);
      for (size_type i = 0; i < N; ++i)
        for (size_type k = 0; k < N; ++k)
          ddi2(i,i,k,k) += scalar_type(1);
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j) {
          ddi2(i,j,j,i) -= scalar_type(1)/scalar_type(2);
          ddi2(j,i,j,i) -= scalar_type(1)/scalar_type(2);
        }
      ddi2_c = true;
    }
 
    inline scalar_type i2()
    { if (!i2_c) compute_i2(); return i2_; }
 
    inline const base_matrix &grad_i2()
    { if (!di2_c) compute_di2(); return di2; }
 
    inline const base_tensor &sym_grad_grad_i2()
    { if (!ddi2_c) compute_ddi2(); return ddi2; }
 
    /* Third invariant det(E) */
    void compute_i3() {
      Einv = E;
      i3_ = bgeot::lu_inverse(&(*(Einv.begin())), gmm::mat_nrows(Einv));
      i3_c = true;
    }
 
    void compute_di3() {
      scalar_type det = i3();
      // gmm::resize(di3, N, N);
      // gmm::copy(gmm::transposed(E), di3);
      di3 = Einv;
      // gmm::lu_inverse(di3);
      gmm::scale(di3, det);
      di3_c = true;
    }
 
    void compute_ddi3() {
      ddi3 = base_tensor(N, N, N, N);
      scalar_type det = i3() / scalar_type(2); // computes also E inverse.
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j)
          for (size_type k = 0; k < N; ++k)
            for (size_type l = 0; l < N; ++l)
              ddi3(i,j,k,l) = det*(Einv(j,i)*Einv(l,k) - Einv(j,k)*Einv(l,i)
                                 + Einv(i,j)*Einv(l,k) - Einv(i,k)*Einv(l,j));
      ddi3_c = true;
    }
 
    inline scalar_type i3()
    { if (!i3_c) compute_i3(); return i3_; }
 
    inline const base_matrix &grad_i3()
    { if (!di3_c) compute_di3(); return di3; }
 
    inline const base_tensor &sym_grad_grad_i3()
    { if (!ddi3_c) compute_ddi3(); return ddi3; }
 
    /* Invariant j1(E) = i1(E)*i3(E)^(-1/3) */
    void compute_j1() {
      j1_ = i1() * ::pow(gmm::abs(i3()), -scalar_type(1) / scalar_type(3));
      j1_c = true;
    }
 
    void compute_dj1() {
      dj1 = grad_i1();
      gmm::add(gmm::scaled(grad_i3(), -i1() / (scalar_type(3) * i3())), dj1);
      gmm::scale(dj1, ::pow(gmm::abs(i3()), -scalar_type(1) / scalar_type(3)));
      dj1_c = true;
    }
 
    void compute_ddj1() {
      const base_matrix &di1_ = grad_i1();
      const base_matrix &di3_ = grad_i3();
      scalar_type coeff1 = scalar_type(1) / (scalar_type(3)*i3());
      scalar_type coeff2 = scalar_type(4) * coeff1 * coeff1 * i1();
      ddj1 = sym_grad_grad_i3();
      gmm::scale(ddj1.as_vector(), -i1() * coeff1);
 
      for (size_type i = 0; i < N; ++i)
         for (size_type j = 0; j < N; ++j)
           for (size_type k = 0; k < N; ++k)
             for (size_type l = 0; l < N; ++l)
              ddj1(i,j,k,l) +=
                (di3_(i, j) * di3_(k, l)) * coeff2
                - (di1_(i, j) * di3_(k, l) + di1_(k, l) * di3_(i, j)) * coeff1;
 
      gmm::scale(ddj1.as_vector(),
                 ::pow(gmm::abs(i3()), -scalar_type(1)/scalar_type(3)));
      ddj1_c = true;
    }
 
    inline scalar_type j1()
    { if (!j1_c) compute_j1(); return j1_; }
 
    inline const base_matrix &grad_j1()
    { if (!dj1_c) compute_dj1(); return dj1; }
 
    inline const base_tensor &sym_grad_grad_j1()
    { if (!ddj1_c) compute_ddj1(); return ddj1; }
 
    /* Invariant j2(E) = i2(E)*i3(E)^(-2/3) */
    void compute_j2() {
      j2_ = i2() * ::pow(gmm::abs(i3()), -scalar_type(2) / scalar_type(3));
      j2_c = true;
    }
 
    void compute_dj2() {
      dj2 = grad_i2();
      gmm::add(gmm::scaled(grad_i3(), -scalar_type(2) * i2() / (scalar_type(3) * i3())), dj2);
      gmm::scale(dj2, ::pow(gmm::abs(i3()), -scalar_type(2) / scalar_type(3)));
      dj2_c = true;
    }
 
    void compute_ddj2() {
      const base_matrix &di2_ = grad_i2();
      const base_matrix &di3_ = grad_i3();
      scalar_type coeff1 = scalar_type(2) / (scalar_type(3)*i3());
      scalar_type coeff2 = scalar_type(5)*coeff1*coeff1*i2() / scalar_type(2);
      ddj2 = sym_grad_grad_i2();
      gmm::add(gmm::scaled(sym_grad_grad_i3().as_vector(), -i2() * coeff1),
               ddj2.as_vector());
 
      for (size_type i = 0; i < N; ++i)
         for (size_type j = 0; j < N; ++j)
           for (size_type k = 0; k < N; ++k)
             for (size_type l = 0; l < N; ++l)
              ddj2(i,j,k,l) +=
                (di3_(i, j) * di3_(k, l)) * coeff2
                - (di2_(i, j) * di3_(k, l) + di2_(k, l) * di3_(i, j)) * coeff1;
 
      gmm::scale(ddj2.as_vector(),
                 ::pow(gmm::abs(i3()), -scalar_type(2)/scalar_type(3)));
      ddj2_c = true;
    }
 
 
    inline scalar_type j2()
    { if (!j2_c) compute_j2(); return j2_; }
 
    inline const base_matrix &grad_j2()
    { if (!dj2_c) compute_dj2(); return dj2; }
 
    inline const base_tensor &sym_grad_grad_j2()
    { if (!ddj2_c) compute_ddj2(); return ddj2; }
 
 
    compute_invariants(const base_matrix &EE)
      : E(EE), i1_c(false), i2_c(false), i3_c(false),
        j1_c(false), j2_c(false), di1_c(false), di2_c(false), di3_c(false),
        dj1_c(false), dj2_c(false), ddi1_c(false), ddi2_c(false),
        ddi3_c(false), ddj1_c(false), ddj2_c(false)
    {
      N = gmm::mat_nrows(E);
      i1_=i2_=i3_=j1_=j2_=0.;
    }
 
  };
 
 
 
 
 
  /* Symmetry check */
 
  int check_symmetry(const base_tensor &t) {
    int flags = 7; size_type N = 3;
    for (size_type n = 0; n < N; ++n)
      for (size_type m = 0; m < N; ++m)
        for (size_type l = 0; l < N; ++l)
          for (size_type k = 0; k < N; ++k) {
            if (gmm::abs(t(n,m,l,k) - t(l,k,n,m))>1e-5) flags &= (~1);
            if (gmm::abs(t(n,m,l,k) - t(m,n,l,k))>1e-5) flags &= (~2);
            if (gmm::abs(t(n,m,l,k) - t(n,m,k,l))>1e-5) flags &= (~4);
          }
    return flags;
  }
 
  /* Member functions of hyperelastic laws */
 
  void abstract_hyperelastic_law::random_E(base_matrix &E) {
    size_type N = gmm::mat_nrows(E);
    base_matrix Phi(N,N);
    scalar_type d;
    do {
      gmm::fill_random(Phi);
      d = bgeot::lu_det(&(*(Phi.begin())), N);
    } while (d < scalar_type(0.01));
    gmm::mult(gmm::transposed(Phi),Phi,E);
    gmm::scale(E,-1.); gmm::add(gmm::identity_matrix(),E);
    gmm::scale(E,-0.5);
  }
 
  void abstract_hyperelastic_law::test_derivatives
  (size_type N, scalar_type h, const base_vector& param) const {
    base_matrix E(N,N), E2(N,N), DE(N,N);
    bool ok = true;
 
    for (size_type count = 0; count < 100; ++count) {
      random_E(E); random_E(DE);
      gmm::scale(DE, h);
      gmm::add(E, DE, E2);
 
      base_matrix sigma1(N,N), sigma2(N,N);
      getfem::base_tensor tdsigma(N,N,N,N);
      base_matrix dsigma(N,N);
      gmm::copy(E, E2); gmm::add(DE, E2);
      sigma(E, sigma1, param, scalar_type(1));
      sigma(E2, sigma2, param, scalar_type(1));
 
      scalar_type d = strain_energy(E2, param, scalar_type(1))
        - strain_energy(E, param, scalar_type(1));
      scalar_type d2 = 0;
      for (size_type i=0; i < N; ++i)
        for (size_type j=0; j < N; ++j) d2 += sigma1(i,j)*DE(i,j);
      if (gmm::abs(d-d2)/(gmm::abs(d)+1e-40) > 1e-4) {
        cout << "Test " << count << " wrong derivative of strain_energy, d="
             << d/h << ", d2=" << d2/h << endl;
        ok = false;
      }
 
      grad_sigma(E,tdsigma,param, scalar_type(1));
      for (size_type i=0; i < N; ++i) {
        for (size_type j=0; j < N; ++j) {
          dsigma(i,j) = 0;
          for (size_type k=0; k < N; ++k) {
            for (size_type m=0; m < N; ++m) {
              dsigma(i,j) += tdsigma(i,j,k,m)*DE(k,m);
            }
          }
          sigma2(i,j) -= sigma1(i,j);
          if (gmm::abs(dsigma(i,j) - sigma2(i,j))
              /(gmm::abs(dsigma(i,j)) + 1e-40) > 1.5e-4) {
            cout << "Test " << count << " wrong derivative of sigma, i="
                 << i << ", j=" << j << ", dsigma=" << dsigma(i,j)/h
                 << ", var sigma = " << sigma2(i,j)/h << endl;
            ok = false;
          }
        }
      }
    }
    GMM_ASSERT1(ok, "Derivative test has failed");
  }
 
  void abstract_hyperelastic_law::cauchy_updated_lagrangian
  (const base_matrix& F, const base_matrix &E,
   base_matrix &cauchy_stress, const base_vector &params,
   scalar_type det_trans) const
  {
    size_type N = E.ncols();
    base_matrix PK2(N,N);
    sigma(E,PK2,params,det_trans);//second Piola-Kirchhoff stress
    base_matrix aux(N,N);
    gmm::mult(F,PK2,aux);
    gmm::mult(aux,gmm::transposed(F),cauchy_stress);
    gmm::scale(cauchy_stress,scalar_type(1.0/det_trans)); //cauchy = 1/J*F*PK2*F^T
  }
 
 
  void abstract_hyperelastic_law::grad_sigma_updated_lagrangian
  (const base_matrix& F, const base_matrix& E,
   const base_vector &params, scalar_type det_trans,
   base_tensor &grad_sigma_ul) const
  {
    size_type N = E.ncols();
    base_tensor Cse(N,N,N,N);
    grad_sigma(E,Cse,params,det_trans);
    scalar_type mult = 1.0/det_trans;
    // this is a general transformation for an anisotropic material, very non-efficient;
    // more effiecient calculations can be overloaded for every specific material
    for(size_type i = 0; i < N; ++i)
      for(size_type j = 0; j < N; ++j)
        for(size_type k = 0; k < N; ++k)
          for(size_type l = 0; l < N; ++l)
            {
              grad_sigma_ul(i,j,k,l) = 0.0;
              for(size_type m = 0; m < N; ++m)
                {    for(size_type n = 0; n < N; ++n)
                    for(size_type p = 0; p < N; ++p)
                      for(size_type q = 0; q < N; ++q)
                        grad_sigma_ul(i,j,k,l)+=
                          F(i,m)*F(j,n)*F(k,p)*F(l,q)*Cse(m,n,p,q);
                }
              grad_sigma_ul(i,j,k,l) *= mult;
            }
  }
 
  scalar_type SaintVenant_Kirchhoff_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params, scalar_type det_trans) const {
        // should be optimized, maybe deriving sigma from strain energy
    if (det_trans <= scalar_type(0))
    { return 1e200; }
 
    return gmm::sqr(gmm::mat_trace(E)) * params[0] / scalar_type(2)
    + gmm::mat_euclidean_norm_sqr(E) * params[1];
  }
 
  void SaintVenant_Kirchhoff_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result,const base_vector &params, scalar_type det_trans) const {
    gmm::copy(gmm::identity_matrix(), result);
    gmm::scale(result, params[0] * gmm::mat_trace(E));
    gmm::add(gmm::scaled(E, 2 * params[1]), result);
    if (det_trans <= scalar_type(0)) {
      gmm::add(gmm::scaled(E, 1e200), result);
    }
  }
  void SaintVenant_Kirchhoff_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,const base_vector &params, scalar_type) const {
    std::fill(result.begin(), result.end(), scalar_type(0));
    size_type N = gmm::mat_nrows(E);
    for (size_type i = 0; i < N; ++i)
      for (size_type l = 0; l < N; ++l) {
        result(i, i, l, l) += params[0];
        result(i, l, i, l) += params[1]/scalar_type(2);
        result(i, l, l, i) += params[1]/scalar_type(2);
        result(l, i, i, l) += params[1]/scalar_type(2);
        result(l, i, l, i) += params[1]/scalar_type(2);
      }
  }
 
  void SaintVenant_Kirchhoff_hyperelastic_law::grad_sigma_updated_lagrangian(const base_matrix& F,
    const base_matrix& E,
    const base_vector &params,
    scalar_type det_trans,
    base_tensor &grad_sigma_ul)const
  {
    size_type N = E.ncols();
    base_tensor Cse(N,N,N,N);
    grad_sigma(E,Cse,params,det_trans);
    base_matrix Cinv(N,N); // left Cauchy-Green deform. tens.
    gmm::mult(F,gmm::transposed(F),Cinv);
    scalar_type mult=1.0/det_trans;
    for(size_type i = 0; i < N; ++i)
      for(size_type j = 0; j < N; ++j)
        for(size_type k = 0; k < N; ++k)
          for(size_type l = 0; l < N; ++l)
            grad_sigma_ul(i, j, k, l)= (Cinv(i,j)*Cinv(k,l)*params[0] +
            params[1]*(Cinv(i,k)*Cinv(j,l) + Cinv(i,l)*Cinv(j,k)))*mult;
  }
 
  SaintVenant_Kirchhoff_hyperelastic_law::SaintVenant_Kirchhoff_hyperelastic_law() {
    nb_params_ = 2;
  }
 
  scalar_type membrane_elastic_law::strain_energy
  (const base_matrix & /* E */, const base_vector & /* params */, scalar_type) const {
    // to be done if needed
    GMM_ASSERT1(false, "To be done");
    return 0;
  }
 
  void membrane_elastic_law::sigma
  (const base_matrix &E, base_matrix &result,const base_vector &params, scalar_type det_trans) const {
    // should be optimized, maybe deriving sigma from strain energy
    base_tensor tt(2,2,2,2);
    size_type N = (gmm::mat_nrows(E) > 2)? 2 : gmm::mat_nrows(E);
    grad_sigma(E,tt,params, det_trans);
    for (size_type i = 0; i < N; ++i)
      for (size_type j = 0; j < N; ++j) {
        result(i,j)=0.0;
        for (size_type k = 0; k < N; ++k)
          for (size_type l = 0; l < N; ++l)
            result(i,j)+=tt(i,j,k,l)*E(k,l);
      }
    // add pretension in X' direction
    if(params[4]!=0) result(0,0)+=params[4];
    // add pretension in Y' direction
    if(params[5]!=0) result(1,1)+=params[5];
    // cout<<"sigma="<<result<<endl;
  }
 
  void membrane_elastic_law::grad_sigma
  (const base_matrix & /* E */, base_tensor &result,
   const base_vector &params, scalar_type) const {
    // to be optimized!!
    std::fill(result.begin(), result.end(), scalar_type(0));
    scalar_type poisonXY=params[0]*params[1]/params[2]; //Ex*vYX=Ey*vXY
    //if no G entered, compute G=E/(2*(1+v))
    scalar_type G=( params[3] == 0) ? params[0]/(2*(1+params[1])) : params[3];
    std::fill(result.begin(), result.end(), scalar_type(0));
    result(0,0,0,0) = params[0]/(1-params[1]*poisonXY);
    // result(0,0,0,1) = 0;
    // result(0,0,1,0) = 0;
    result(0,0,1,1) = params[1]*params[0]/(1-params[1]*poisonXY);
    result(1,1,0,0) = params[1]*params[0]/(1-params[1]*poisonXY);
    // result(1,1,0,1) = 0;
    // result(1,1,1,0) = 0;
    result(1,1,1,1) = params[2]/(1-params[1]*poisonXY);
    // result(0,1,0,0) = 0;
    result(0,1,0,1) = G;
    result(0,1,1,0) = G;
    // result(0,1,1,1) = 0;
    // result(1,0,0,0) = 0;
    result(1,0,0,1) = G;
    result(1,0,1,0) = G;
    // result(1,0,1,1) = 0;
  }
 
  scalar_type Mooney_Rivlin_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params
   ,scalar_type  det_trans) const {
    //shouldn't negative det_trans be handled here???
    if (compressible && det_trans <= scalar_type(0)) return 1e200;
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Mooney Rivlin hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
    size_type i=0;
    scalar_type C1 = params[i++]; // C10
    scalar_type W = C1 * (ci.j1() - scalar_type(3));
    if (!neohookean) {
      scalar_type C2 = params[i++]; // C01
      W += C2 * (ci.j2() - scalar_type(3));
    }
    if (compressible) {
      scalar_type D1 = params[i++];
      W += D1 * gmm::sqr(sqrt(gmm::abs(ci.i3())) - scalar_type(1));
    }
    return W;
  }
 
  void Mooney_Rivlin_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result,
   const base_vector &params, scalar_type det_trans) const {
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Mooney Rivlin hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    size_type i=0;
    scalar_type C1 = params[i++]; // C10
    gmm::copy(gmm::scaled(ci.grad_j1(), scalar_type(2) * C1), result);
    if (!neohookean) {
      scalar_type C2 = params[i++]; // C01
      gmm::add(gmm::scaled(ci.grad_j2(), scalar_type(2) * C2), result);
    }
    if (compressible) {
      scalar_type D1 = params[i++];
      scalar_type di3 = D1 - D1 / sqrt(gmm::abs(ci.i3()));
      gmm::add(gmm::scaled(ci.grad_i3(), scalar_type(2) * di3), result);
 
// shouldn't negative det_trans be handled here???
      if (det_trans <= scalar_type(0))
        gmm::add(gmm::scaled(C, 1e200), result);
    }
  }
 
  void Mooney_Rivlin_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,
   const base_vector &params, scalar_type) const {
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Mooney Rivlin hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    size_type i=0;
    scalar_type C1 = params[i++]; // C10
    gmm::copy(gmm::scaled(ci.sym_grad_grad_j1().as_vector(),
                          scalar_type(4)*C1), result.as_vector());
    if (!neohookean) {
      scalar_type C2 = params[i++]; // C01
      gmm::add(gmm::scaled(ci.sym_grad_grad_j2().as_vector(),
               scalar_type(4)*C2), result.as_vector());
    }
    if (compressible) {
      scalar_type D1 = params[i++];
      scalar_type di3 = D1 - D1 / sqrt(gmm::abs(ci.i3()));
      gmm::add(gmm::scaled(ci.sym_grad_grad_i3().as_vector(),
                               scalar_type(4)*di3), result.as_vector());
 
      // second derivatives of W with respect to the third invariant
      scalar_type A22 = D1 / (scalar_type(2) * pow(gmm::abs(ci.i3()), 1.5));
      const base_matrix &di = ci.grad_i3();
      for (size_type l1 = 0; l1 < N; ++l1)
        for (size_type l2 = 0; l2 < N; ++l2)
          for (size_type l3 = 0; l3 < N; ++l3)
            for (size_type l4 = 0; l4 < N; ++l4)
              result(l1, l2, l3, l4) +=
                scalar_type(4) * A22 * di(l1, l2) * di(l3, l4);
    }
 
//     GMM_ASSERT1(check_symmetry(result) == 7,
//                 "Fourth order tensor not symmetric : " << result);
  }
 
  Mooney_Rivlin_hyperelastic_law::Mooney_Rivlin_hyperelastic_law
  (bool compressible_, bool neohookean_)
  : compressible(compressible_), neohookean(neohookean_)
  {
    nb_params_ = 2;
    if (compressible) ++nb_params_; // D1 != 0
    if (neohookean) --nb_params_;   // C2 == 0
 
  }
 
 
 
 
  scalar_type Neo_Hookean_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params, scalar_type det_trans) const {
   if (det_trans <= scalar_type(0)) return 1e200;
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Neo Hookean hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    scalar_type lambda = params[0];
    scalar_type mu = params[1];
    scalar_type logi3 = log(ci.i3());
    scalar_type W = mu/2 * (ci.i1() - scalar_type(3) - logi3);
    if (bonet)
      W += lambda/8 * gmm::sqr(logi3);
    else // Wriggers
      W += lambda/4 * (ci.i3() - scalar_type(1) - logi3);
 
    return W;
  }
 
  void Neo_Hookean_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result,
   const base_vector &params , scalar_type det_trans) const {
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Neo Hookean hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    scalar_type lambda = params[0];
    scalar_type mu = params[1];
    gmm::copy(gmm::scaled(ci.grad_i1(), mu), result);
    if (bonet)
       gmm::add(gmm::scaled(ci.grad_i3(),
                            (lambda/2 * log(ci.i3()) - mu) / ci.i3()), result);
    else
       gmm::add(gmm::scaled(ci.grad_i3(),
                            lambda/2 - lambda/(2*ci.i3()) - mu / ci.i3()), result);
    if (det_trans <= scalar_type(0))
      gmm::add(gmm::scaled(C, 1e200), result);
  }
 
  void Neo_Hookean_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,
   const base_vector &params, scalar_type) const {
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Neo Hookean hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    scalar_type lambda = params[0];
    scalar_type mu = params[1];
 
    scalar_type coeff;
    if (bonet) {
      scalar_type logi3 = log(ci.i3());
      gmm::copy(gmm::scaled(ci.sym_grad_grad_i3().as_vector(),
                            (lambda * logi3 - 2*mu) / ci.i3()),
                result.as_vector());
      coeff = (lambda + 2 * mu - lambda * logi3) / gmm::sqr(ci.i3());
    } else {
      gmm::copy(gmm::scaled(ci.sym_grad_grad_i3().as_vector(),
                            lambda  - (lambda + 2 * mu) / ci.i3()),
                result.as_vector());
      coeff = (lambda + 2 * mu) / gmm::sqr(ci.i3());
    }
 
    const base_matrix &di = ci.grad_i3();
    for (size_type l1 = 0; l1 < N; ++l1)
      for (size_type l2 = 0; l2 < N; ++l2)
        for (size_type l3 = 0; l3 < N; ++l3)
          for (size_type l4 = 0; l4 < N; ++l4)
            result(l1, l2, l3, l4) += coeff * di(l1, l2) * di(l3, l4);
 
//     GMM_ASSERT1(check_symmetry(result) == 7,
//                 "Fourth order tensor not symmetric : " << result);
  }
 
  Neo_Hookean_hyperelastic_law::Neo_Hookean_hyperelastic_law(bool bonet_)
    : bonet(bonet_)
  {
    nb_params_ = 2;
  }
 
 
 
  scalar_type generalized_Blatz_Ko_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params, scalar_type det_trans) const {
    if (det_trans <= scalar_type(0)) return 1e200;
    scalar_type a = params[0], b = params[1], c = params[2], d = params[3];
    scalar_type n = params[4];
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Generalized Blatz Ko hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    return pow(a*ci.i1() + b*sqrt(gmm::abs(ci.i3()))
               + c*ci.i2() / ci.i3() + d, n);
  }
 
  void generalized_Blatz_Ko_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result,
   const base_vector &params, scalar_type det_trans) const {
    scalar_type a = params[0], b = params[1], c = params[2], d = params[3];
    scalar_type n = params[4];
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Generalized Blatz Ko hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
    scalar_type z = a*ci.i1() + b*sqrt(gmm::abs(ci.i3()))
      + c*ci.i2() / ci.i3() + d;
    scalar_type nz = n * pow(z, n-1.);
    scalar_type di1 = nz * a;
    scalar_type di2 = nz * c / ci.i3();
    scalar_type di3 = nz *
      (b / (2. * sqrt(gmm::abs(ci.i3()))) - c * ci.i2() / gmm::sqr(ci.i3()));
 
    gmm::copy(gmm::scaled(ci.grad_i1(), di1 * 2.0), result);
    gmm::add(gmm::scaled(ci.grad_i2(), di2 * 2.0), result);
    gmm::add(gmm::scaled(ci.grad_i3(), di3 * 2.0), result);
    if (det_trans <= scalar_type(0))
      gmm::add(gmm::scaled(C, 1e200), result);
 
  }
 
  void generalized_Blatz_Ko_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,
   const base_vector &params, scalar_type) const {
    scalar_type a = params[0], b = params[1], c = params[2], d = params[3];
    scalar_type n = params[4];
    size_type N = gmm::mat_nrows(E);
    GMM_ASSERT1(N == 3, "Generalized Blatz Ko hyperelastic law only defined "
                "on dimension 3, sorry");
    base_matrix C = E;
    gmm::scale(C, scalar_type(2));
    gmm::add(gmm::identity_matrix(), C);
    compute_invariants ci(C);
 
 
    scalar_type z = a*ci.i1() + b*sqrt(gmm::abs(ci.i3()))
      + c*ci.i2() / ci.i3() + d;
    scalar_type nz = n * pow(z, n-1.);
    scalar_type di1 = nz * a;
    scalar_type di2 = nz * c / ci.i3();
    scalar_type y = (b / (2. * sqrt(gmm::abs(ci.i3()))) - c * ci.i2() / gmm::sqr(ci.i3()));
    scalar_type di3 = nz * y;
 
    gmm::copy(gmm::scaled(ci.sym_grad_grad_i1().as_vector(),
                          scalar_type(4)*di1), result.as_vector());
    gmm::add(gmm::scaled(ci.sym_grad_grad_i2().as_vector(),
                         scalar_type(4)*di2), result.as_vector());
    gmm::add(gmm::scaled(ci.sym_grad_grad_i3().as_vector(),
                         scalar_type(4)*di3), result.as_vector());
 
    scalar_type nnz = n * (n-1.) * pow(z, n-2.);
    base_matrix A(3, 3); // second derivatives of W with respect to invariants
    A(0, 0) = nnz * a * a;
    A(1, 0) = A(0, 1) = nnz * a * c / ci.i3();
    A(2, 0) = A(0, 2) = nnz * a * y;
    A(1, 1) = nnz * c * c / gmm::sqr(ci.i3());
    A(2, 1) = A(1, 2) = nnz * y * c / ci.i3() - nz * c / gmm::sqr(ci.i3());
    A(2, 2) = nnz * y * y + nz * (2. * c * ci.i2() / pow(ci.i3(), 3.) - b / (4. * pow(ci.i3(), 1.5)));
 
    typedef const base_matrix * pointer_base_matrix__;
    pointer_base_matrix__ di[3];
    di[0] = &(ci.grad_i1());
    di[1] = &(ci.grad_i2());
    di[2] = &(ci.grad_i3());
 
    for (size_type j = 0; j < N; ++j)
      for (size_type k = 0; k < N; ++k) {
        for (size_type l1 = 0; l1 < N; ++l1)
          for (size_type l2 = 0; l2 < N; ++l2)
            for (size_type l3 = 0; l3 < N; ++l3)
              for (size_type l4 = 0; l4 < N; ++l4)
                result(l1, l2, l3, l4)
                  += 4. * A(j, k) * (*di[j])(l1, l2) * (*di[k])(l3, l4);
      }
 
//     GMM_ASSERT1(check_symmetry(result) == 7,
//                 "Fourth order tensor not symmetric : " << result);
  }
 
  generalized_Blatz_Ko_hyperelastic_law::generalized_Blatz_Ko_hyperelastic_law() {
    nb_params_ = 5;
    base_vector V(5);
    V[0] = 1.0;  V[1] = 1.0, V[2] = 1.5; V[3] = -0.5; V[4] = 1.5;
  }
 
 
  scalar_type Ciarlet_Geymonat_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params, scalar_type det_trans) const {
    if (det_trans <= scalar_type(0)) return 1e200;
    size_type N = gmm::mat_nrows(E);
    scalar_type a = params[2];
    scalar_type b = params[1]/scalar_type(2) - params[2];
    scalar_type c = params[0]/scalar_type(4) - params[1]/scalar_type(2)
                    + params[2];
    scalar_type d = params[0]/scalar_type(2) + params[1];
    scalar_type e = -(scalar_type(3)*(a+b) + c);
    base_matrix C(N, N);
    gmm::copy(gmm::scaled(E, scalar_type(2)), C);
    gmm::add(gmm::identity_matrix(), C);
    scalar_type det = bgeot::lu_det(&(*(C.begin())), N);
    return a * gmm::mat_trace(C)
      + b * (gmm::sqr(gmm::mat_trace(C)) -
             gmm::mat_euclidean_norm_sqr(C))/scalar_type(2)
      + c * det - d * log(det) / scalar_type(2) + e;
  }
 
  void Ciarlet_Geymonat_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result, const base_vector &params, scalar_type det_trans) const {
    size_type N = gmm::mat_nrows(E);
    scalar_type a = params[2];
    scalar_type b = params[1]/scalar_type(2) - params[2];
    scalar_type c = params[0]/scalar_type(4) - params[1]/scalar_type(2)
                    + params[2];
    scalar_type d = params[0]/scalar_type(2) + params[1];
    base_matrix C(N, N);
    if (a > params[1]/scalar_type(2)
        || a < params[1]/scalar_type(2) - params[0]/scalar_type(4) || a < 0)
      GMM_WARNING1("Inconsistent third parameter for Ciarlet-Geymonat "
                   "hyperelastic law");
    gmm::copy(gmm::scaled(E, scalar_type(2)), C);
    gmm::add(gmm::identity_matrix(), C);
    gmm::copy(gmm::identity_matrix(), result);
    gmm::scale(result, scalar_type(2) * (a + b * gmm::mat_trace(C)));
    gmm::add(gmm::scaled(C, -scalar_type(2) * b), result);
    if (det_trans <= scalar_type(0))
      gmm::add(gmm::scaled(C, 1e200), result);
    else {
      scalar_type det = bgeot::lu_inverse(&(*(C.begin())), N);
      gmm::add(gmm::scaled(C, scalar_type(2) * c * det - d), result);
    }
  }
 
  void Ciarlet_Geymonat_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,const base_vector &params, scalar_type) const {
    size_type N = gmm::mat_nrows(E);
    // scalar_type a = params[2];
    scalar_type b2 = params[1] - params[2]*scalar_type(2); // b*2
    scalar_type c = params[0]/scalar_type(4) - params[1]/scalar_type(2)
                    + params[2];
    scalar_type d = params[0]/scalar_type(2) + params[1];
    base_matrix C(N, N);
    gmm::copy(gmm::scaled(E, scalar_type(2)), C);
    gmm::add(gmm::identity_matrix(), C);
    scalar_type det = bgeot::lu_inverse(&(*(C.begin())), N);
    std::fill(result.begin(), result.end(), scalar_type(0));
    for (size_type i = 0; i < N; ++i)
      for (size_type j = 0; j < N; ++j) {
        result(i, i, j, j) += 2*b2;
        result(i, j, i, j) -= b2;
        result(i, j, j, i) -= b2;
        for (size_type  k = 0; k < N; ++k)
          for (size_type  l = 0; l < N; ++l)
            result(i, j, k, l) +=
              (C(i, k)*C(l, j) + C(i, l)*C(k, j)) * (d-scalar_type(2)*det*c)
              + (C(i, j) * C(k, l)) * det*c*scalar_type(4);
      }
 
//     GMM_ASSERT1(check_symmetry(result) == 7,
//                 "Fourth order tensor not symmetric : " << result);
  }
 
 
  int levi_civita(int i, int j, int k) {
    int ii=i+1;
    int jj=j+1;
    int kk=k+1; //i,j,k from 0 to 2 !
    return static_cast<int>
      (int(- 1)*(static_cast<int>(pow(double(ii-jj),2.))%3)
       * (static_cast<int> (pow(double(ii-kk),double(2)))%3 )
       * (static_cast<int> (pow(double(jj-kk),double(2)))%3)
       * (pow(double(jj-(ii%3))-double(0.5),double(2))-double(1.25)));
  }
 
 
 
  scalar_type plane_strain_hyperelastic_law::strain_energy
  (const base_matrix &E, const base_vector &params, scalar_type det_trans) const {
    GMM_ASSERT1(gmm::mat_nrows(E) == 2, "Plane strain law is for 2D only.");
    base_matrix E3D(3,3);
    E3D(0,0)=E(0,0); E3D(1,0)=E(1,0); E3D(0,1)=E(0,1); E3D(1,1)=E(1,1);
    return pl->strain_energy(E3D, params, det_trans);
  }
 
  void plane_strain_hyperelastic_law::sigma
  (const base_matrix &E, base_matrix &result,const base_vector &params, scalar_type det_trans) const {
    GMM_ASSERT1(gmm::mat_nrows(E) == 2, "Plane strain law is for 2D only.");
    base_matrix E3D(3,3), result3D(3,3);
    E3D(0,0)=E(0,0); E3D(1,0)=E(1,0); E3D(0,1)=E(0,1); E3D(1,1)=E(1,1);
    pl->sigma(E3D, result3D, params, det_trans);
    result(0,0) = result3D(0,0); result(1,0) = result3D(1,0);
    result(0,1) = result3D(0,1); result(1,1) = result3D(1,1);
  }
 
  void plane_strain_hyperelastic_law::grad_sigma
  (const base_matrix &E, base_tensor &result,const base_vector &params, scalar_type det_trans) const {
    GMM_ASSERT1(gmm::mat_nrows(E) == 2, "Plane strain law is for 2D only.");
    base_matrix E3D(3,3);
    base_tensor result3D(3,3,3,3);
    E3D(0,0)=E(0,0); E3D(1,0)=E(1,0); E3D(0,1)=E(0,1); E3D(1,1)=E(1,1);
    pl->grad_sigma(E3D, result3D, params, det_trans);
    result(0,0,0,0) = result3D(0,0,0,0); result(1,0,0,0) = result3D(1,0,0,0);
    result(0,1,0,0) = result3D(0,1,0,0); result(1,1,0,0) = result3D(1,1,0,0);
    result(0,0,1,0) = result3D(0,0,1,0); result(1,0,1,0) = result3D(1,0,1,0);
    result(0,1,1,0) = result3D(0,1,1,0); result(1,1,1,0) = result3D(1,1,1,0);
    result(0,0,0,1) = result3D(0,0,0,1); result(1,0,0,1) = result3D(1,0,0,1);
    result(0,1,0,1) = result3D(0,1,0,1); result(1,1,0,1) = result3D(1,1,0,1);
    result(0,0,1,1) = result3D(0,0,1,1); result(1,0,1,1) = result3D(1,0,1,1);
    result(0,1,1,1) = result3D(0,1,1,1); result(1,1,1,1) = result3D(1,1,1,1);
  }
 
 
 
 
 
 
 
  //=========================================================================
  //
  //  Nonlinear elasticity (old) Brick
  //
  //=========================================================================
 
  struct nonlinear_elasticity_brick : public virtual_brick {
 
    phyperelastic_law AHL;
 
    virtual void asm_real_tangent_terms(const model &md, size_type /* ib */,
                                        const model::varnamelist &vl,
                                        const model::varnamelist &dl,
                                        const model::mimlist &mims,
                                        model::real_matlist &matl,
                                        model::real_veclist &vecl,
                                        model::real_veclist &,
                                        size_type region,
                                        build_version version) const {
      GMM_ASSERT1(mims.size() == 1,
                  "Nonlinear elasticity brick need a single mesh_im");
      GMM_ASSERT1(vl.size() == 1,
                  "Nonlinear elasticity brick need a single variable");
      GMM_ASSERT1(dl.size() == 1,
                  "Wrong number of data for nonlinear elasticity brick, "
                  << dl.size() << " should be 1 (vector).");
      GMM_ASSERT1(matl.size() == 1,  "Wrong number of terms for nonlinear "
                  "elasticity brick");
 
      const model_real_plain_vector &u = md.real_variable(vl[0]);
      const mesh_fem &mf_u = *(md.pmesh_fem_of_variable(vl[0]));
 
      const mesh_fem *mf_params = md.pmesh_fem_of_variable(dl[0]);
      const model_real_plain_vector &params = md.real_variable(dl[0]);
      const mesh_im &mim = *mims[0];
 
      size_type sl = gmm::vect_size(params);
      if (mf_params) sl = sl * mf_params->get_qdim() / mf_params->nb_dof();
      GMM_ASSERT1(sl == AHL->nb_params(), "Wrong number of coefficients for the "
                  "nonlinear constitutive elastic law");
 
      mesh_region rg(region);
      mf_u.linked_mesh().intersect_with_mpi_region(rg);
 
      if (version & model::BUILD_MATRIX) {
        gmm::clear(matl[0]);
        GMM_TRACE2("Nonlinear elasticity stiffness matrix assembly");
        asm_nonlinear_elasticity_tangent_matrix
          (matl[0], mim, mf_u, u, mf_params, params, *AHL, rg);
      }
 
 
      if (version & model::BUILD_RHS) {
        asm_nonlinear_elasticity_rhs(vecl[0], mim,
                                     mf_u, u, mf_params, params, *AHL, rg);
        gmm::scale(vecl[0], scalar_type(-1));
      }
 
    }
 
    nonlinear_elasticity_brick(const phyperelastic_law &AHL_)
      : AHL(AHL_) {
      set_flags("Nonlinear elasticity brick", false /* is linear*/,
                true /* is symmetric */, true /* is coercive */,
                true /* is real */, false /* is complex */);
    }
 
  };
 
  //=========================================================================
  //  Add a nonlinear elasticity brick.
  //=========================================================================
 
  // Deprecated brick
  size_type add_nonlinear_elasticity_brick
  (model &md, const mesh_im &mim, const std::string &varname,
   const phyperelastic_law &AHL, const std::string &dataname,
   size_type region) {
    pbrick pbr = std::make_shared<nonlinear_elasticity_brick>(AHL);
 
    model::termlist tl;
    tl.push_back(model::term_description(varname, varname, true));
    model::varnamelist dl(1, dataname);
    model::varnamelist vl(1, varname);
    return md.add_brick(pbr, vl, dl, tl, model::mimlist(1,&mim), region);
  }
 
  //=========================================================================
  //  Von Mises or Tresca stress computation.
  //=========================================================================
 
  void compute_Von_Mises_or_Tresca(model &md,
                                   const std::string &varname,
                                   const phyperelastic_law &AHL,
                                   const std::string &dataname,
                                   const mesh_fem &mf_vm,
                                   model_real_plain_vector &VM,
                                   bool tresca) {
    GMM_ASSERT1(gmm::vect_size(VM) == mf_vm.nb_dof(),
                "The vector has not the good size");
    const mesh_fem &mf_u = md.mesh_fem_of_variable(varname);
    const model_real_plain_vector &u = md.real_variable(varname);
    const mesh_fem *mf_params = md.pmesh_fem_of_variable(dataname);
    const model_real_plain_vector &params = md.real_variable(dataname);
 
    size_type sl = gmm::vect_size(params);
    if (mf_params) sl = sl * mf_params->get_qdim() / mf_params->nb_dof();
    GMM_ASSERT1(sl == AHL->nb_params(), "Wrong number of coefficients for "
                "the nonlinear constitutive elastic law");
 
    unsigned N = unsigned(mf_u.linked_mesh().dim());
    unsigned NP = unsigned(AHL->nb_params()), NFem = mf_u.get_qdim();
    model_real_plain_vector GRAD(mf_vm.nb_dof()*NFem*N);
    model_real_plain_vector PARAMS(mf_vm.nb_dof()*NP);
    if (mf_params) interpolation(*mf_params, mf_vm, params, PARAMS);
    compute_gradient(mf_u, mf_vm, u, GRAD);
    base_matrix E(N, N), gradphi(NFem,N),gradphit(N,NFem), Id(N, N),
      sigmahathat(N,N),aux(NFem,N), sigma(NFem,NFem),
      IdNFem(NFem, NFem);
    base_vector p(NP);
    if (!mf_params) gmm::copy(params, p);
    base_vector eig(NFem);
    base_vector ez(NFem);    // vector normal at deformed surface, (ex X ey)
    double normEz(0);        //norm of ez
    gmm::copy(gmm::identity_matrix(), Id);
    gmm::copy(gmm::identity_matrix(), IdNFem);
    for (size_type i = 0; i < mf_vm.nb_dof(); ++i) {
      gmm::resize(gradphi,NFem,N);
      std::copy(GRAD.begin()+i*NFem*N, GRAD.begin()+(i+1)*NFem*N,
                gradphit.begin());
      gmm::copy(gmm::transposed(gradphit),gradphi);
      for (unsigned int alpha = 0; alpha <N; ++alpha)
        gradphi(alpha, alpha)+=1;
      gmm::mult(gmm::transposed(gradphi), gradphi, E);
      gmm::add(gmm::scaled(Id, -scalar_type(1)), E);
      gmm::scale(E, scalar_type(1)/scalar_type(2));
      if (mf_params)
        gmm::copy(gmm::sub_vector(PARAMS, gmm::sub_interval(i*NP,NP)), p);
      AHL->sigma(E, sigmahathat, p, scalar_type(1));
      if (NFem == 3 && N == 2) {
        //jyh : compute ez, normal on deformed surface
        for (unsigned int l = 0; l <NFem; ++l)  {
          ez[l]=0;
          for (unsigned int m = 0; m <NFem; ++m)
            for (unsigned int n = 0; n <NFem; ++n){
              ez[l]+=levi_civita(l,m,n)*gradphi(m,0)*gradphi(n,1);
            }
          normEz= gmm::vect_norm2(ez);
        }
        //jyh : end compute ez
      }
      gmm::mult(gradphi, sigmahathat, aux);
      gmm::mult(aux, gmm::transposed(gradphi), sigma);
 
      /* jyh : complete gradphi for virtual 3rd dim (perpendicular to
         deformed surface, same thickness) */
      if (NFem == 3 && N == 2) {
        gmm::resize(gradphi,NFem,NFem);
        for (unsigned int ll = 0; ll <NFem; ++ll)
          for (unsigned int ii = 0; ii <NFem; ++ii)
            for (unsigned int jj = 0; jj <NFem; ++jj)
              gradphi(ll,2)+=(levi_civita(ll,ii,jj)*gradphi(ii,0)
                              *gradphi(jj,1))/normEz;
        //jyh : end complete graphi
      }
 
      gmm::scale(sigma, scalar_type(1) / bgeot::lu_det(&(*(gradphi.begin())), NFem));
 
      if (!tresca) {
        /* von mises: norm(deviator(sigma)) */
        gmm::add(gmm::scaled(IdNFem, -gmm::mat_trace(sigma) / NFem), sigma);
 
        //jyh : von mises stress=sqrt(3/2)* norm(sigma) ?
        VM[i] = sqrt(3.0/2)*gmm::mat_euclidean_norm(sigma);
      } else {
        /* else compute the tresca criterion */
        //jyh : to be adapted for membrane if necessary
        gmm::symmetric_qr_algorithm(sigma, eig);
        std::sort(eig.begin(), eig.end());
        VM[i] = eig.back() - eig.front();
      }
    }
  }
 
 
  void compute_sigmahathat(model &md,
                           const std::string &varname,
                           const phyperelastic_law &AHL,
                           const std::string &dataname,
                           const mesh_fem &mf_sigma,
                           model_real_plain_vector &SIGMA) {
    const mesh_fem &mf_u = md.mesh_fem_of_variable(varname);
    const model_real_plain_vector &u = md.real_variable(varname);
    const mesh_fem *mf_params = md.pmesh_fem_of_variable(dataname);
    const model_real_plain_vector &params = md.real_variable(dataname);
 
    size_type sl = gmm::vect_size(params);
    if (mf_params) sl = sl * mf_params->get_qdim() / mf_params->nb_dof();
    GMM_ASSERT1(sl == AHL->nb_params(), "Wrong number of coefficients for "
                "the nonlinear constitutive elastic law");
 
    unsigned N = unsigned(mf_u.linked_mesh().dim());
    unsigned NP = unsigned(AHL->nb_params()), NFem = mf_u.get_qdim();
    GMM_ASSERT1(mf_sigma.nb_dof() > 0, "Bad mf_sigma");
    size_type qqdim = mf_sigma.get_qdim();
    size_type ratio = N*N / qqdim;
 
    GMM_ASSERT1(((ratio == 1) || (ratio == N*N)) &&
                (gmm::vect_size(SIGMA) == mf_sigma.nb_dof()*ratio),
                "The vector has not the good size");
 
    model_real_plain_vector GRAD(mf_sigma.nb_dof()*ratio*NFem/N);
    model_real_plain_vector PARAMS(mf_sigma.nb_dof()*NP);
 
 
    getfem::mesh_trans_inv mti(mf_sigma.linked_mesh());
    if (mf_params) {
      for (size_type i = 0; i < mf_sigma.nb_dof(); ++i)
        mti.add_point(mf_sigma.point_of_basic_dof(i));
      interpolation(*mf_params, mti, params, PARAMS);
    }
 
    compute_gradient(mf_u, mf_sigma, u, GRAD);
    base_matrix E(N, N), gradphi(NFem,N),gradphit(N,NFem), Id(N, N),
      sigmahathat(N,N),aux(NFem,N), sigma(NFem,NFem),
      IdNFem(NFem, NFem);
 
 
    base_vector p(NP);
    if (!mf_params) gmm::copy(params, p);
    base_vector eig(NFem);
    base_vector ez(NFem); // vector normal at deformed surface, (ex X ey)
    gmm::copy(gmm::identity_matrix(), Id);
    gmm::copy(gmm::identity_matrix(), IdNFem);
 
    // cout << "GRAD = " << GRAD << endl;
    // GMM_ASSERT1(false, "oops");
 
    for (size_type i = 0; i < mf_sigma.nb_dof()/qqdim; ++i) {
//       cout << "GRAD size = " << gmm::vect_size(GRAD) << endl;
//       cout << "i = " << i << endl;
//       cout << "i*NFem*N = " << i*NFem*N << endl;
 
      gmm::resize(gradphi,NFem,N);
      std::copy(GRAD.begin()+i*NFem*N, GRAD.begin()+(i+1)*NFem*N,
                gradphit.begin());
      // cout << "GRAD = " << gradphit << endl;
      gmm::copy(gmm::transposed(gradphit),gradphi);
      for (unsigned int alpha = 0; alpha <N; ++alpha)
        gradphi(alpha, alpha) += scalar_type(1);
      gmm::mult(gmm::transposed(gradphi), gradphi, E);
      gmm::add(gmm::scaled(Id, -scalar_type(1)), E);
      gmm::scale(E, scalar_type(1)/scalar_type(2));
      if (mf_params)
        gmm::copy(gmm::sub_vector(PARAMS, gmm::sub_interval(i*ratio*NP,NP)),p);
      // cout << "E = " << E << endl;
      AHL->sigma(E, sigmahathat, p, scalar_type(1));
      // cout << "ok" << endl;
      std::copy(sigmahathat.begin(), sigmahathat.end(), SIGMA.begin()+i*N*N);
    }
  }
 
 
 
  // ----------------------------------------------------------------------
  //
  // Nonlinear incompressibility brick
  //
  // ----------------------------------------------------------------------
 
  struct nonlinear_incompressibility_brick : public virtual_brick {
 
    virtual void asm_real_tangent_terms(const model &md, size_type,
                                        const model::varnamelist &vl,
                                        const model::varnamelist &dl,
                                        const model::mimlist &mims,
                                        model::real_matlist &matl,
                                        model::real_veclist &vecl,
                                        model::real_veclist &veclsym,
                                        size_type region,
                                        build_version version) const {
 
      GMM_ASSERT1(matl.size() == 2,  "Wrong number of terms for nonlinear "
                  "incompressibility brick");
      GMM_ASSERT1(dl.size() == 0, "Nonlinear incompressibility brick need no "
                  "data");
      GMM_ASSERT1(mims.size() == 1, "Nonlinear incompressibility brick need a "
                  "single mesh_im");
      GMM_ASSERT1(vl.size() == 2, "Wrong number of variables for nonlinear "
                  "incompressibility brick");
 
      const mesh_fem &mf_u = md.mesh_fem_of_variable(vl[0]);
      const mesh_fem &mf_p = md.mesh_fem_of_variable(vl[1]);
      const model_real_plain_vector &u = md.real_variable(vl[0]);
      const model_real_plain_vector &p = md.real_variable(vl[1]);
      const mesh_im &mim = *mims[0];
      mesh_region rg(region);
      mim.linked_mesh().intersect_with_mpi_region(rg);
 
      if (version & model::BUILD_MATRIX) {
        gmm::clear(matl[0]);
        gmm::clear(matl[1]);
        asm_nonlinear_incomp_tangent_matrix(matl[0], matl[1],
                                            mim, mf_u, mf_p, u, p, rg);
      }
 
      if (version & model::BUILD_RHS) {
        asm_nonlinear_incomp_rhs(vecl[0], veclsym[1], mim, mf_u, mf_p,u,p, rg);
        gmm::scale(vecl[0], scalar_type(-1));
        gmm::scale(veclsym[1], scalar_type(-1));
      }
    }
 
 
    nonlinear_incompressibility_brick() {
      set_flags("Nonlinear incompressibility brick",
                false /* is linear*/,
                true /* is symmetric */, false /* is coercive */,
                true /* is real */, false /* is complex */);
    }
 
 
  };
 
  size_type add_nonlinear_incompressibility_brick
  (model &md, const mesh_im &mim, const std::string &varname,
   const std::string &multname, size_type region) {
    pbrick pbr = std::make_shared<nonlinear_incompressibility_brick>();
    model::termlist tl;
    tl.push_back(model::term_description(varname, varname, true));
    tl.push_back(model::term_description(varname, multname, true));
    model::varnamelist vl(1, varname);
    vl.push_back(multname);
    model::varnamelist dl;
    return md.add_brick(pbr, vl, dl, tl, model::mimlist(1, &mim), region);
  }
 
 
 
 
 
  // ----------------------------------------------------------------------
  //
  // High-level Generic assembly operators
  //
  // ----------------------------------------------------------------------
 
 
  static void ga_init_scalar_(bgeot::multi_index &mi) { mi.resize(0); }
  static void ga_init_square_matrix_(bgeot::multi_index &mi, size_type N)
  { mi.resize(2); mi[0] = mi[1] = N; }
 
 
  // Matrix_i2 Operator (second invariant of square matrix of size >= 2)
  //                    (For 2x2 matrices, it is equivalent to det(M))
  struct matrix_i2_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_scalar_(sizes);
      return true;
    }
 
    // Value : (Trace(M))^2 - Trace(M^2))/2
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      const base_tensor &t = *args[0];
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += t[i*N+i];
      scalar_type tr2 = scalar_type(0);
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j)
          tr2 += t[i+ j*N] * t[j + i*N];
      result[0] = (tr*tr-tr2)/scalar_type(2);
    }
 
    // Derivative : Trace(M)I - M^T
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      const base_tensor &t = *args[0];
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += t[i*N+i];
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < N; ++j)
        for (size_type i = 0; i < N; ++i, ++it)
          *it = ((i == j) ? tr : scalar_type(0)) - t[i*N+j];
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative : I@I - \delta_{il}\delta_{jk}
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      gmm::clear(result.as_vector());
       for (size_type i = 0; i < N; ++i)
         for (size_type j = 0; j < N; ++j) {
           result[(N+1)*(i+N*N*j)] += scalar_type(1);
           result[(N+1)*N*j + i*(N*N*N + 1)] -= scalar_type(1);
         }
    }
  };
 
 
  // Matrix_j1 Operator
  struct matrix_j1_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_scalar_(sizes);
      return true;
    }
 
    // Value : Trace(M)/(det(M)^1/3)
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_matrix M(N, N);
      gmm::copy(args[0]->as_vector(), M.as_vector());
      scalar_type det = bgeot::lu_det(&(*(M.begin())), N);
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      if (det > 0)
        result[0] = tr / pow(det, scalar_type(1)/scalar_type(3));
      else
        result[0] = 1.E200;
    }
 
    // Derivative : (I-Trace(M)*M^{-T}/3)/(det(M)^1/3)
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_matrix M(N, N);
      gmm::copy(args[0]->as_vector(), M.as_vector());
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      scalar_type det = bgeot::lu_inverse(&(*(M.begin())), N);
      if (det > 0) {
        base_tensor::iterator it = result.begin();
        for (size_type j = 0; j < N; ++j)
          for (size_type i = 0; i < N; ++i, ++it)
            *it = (((i == j) ? scalar_type(1) : scalar_type(0))
                   - tr*M(j,i)/scalar_type(3))
              / pow(det, scalar_type(1)/scalar_type(3));
        GMM_ASSERT1(it == result.end(), "Internal error");
      } else
        std::fill(result.begin(), result.end(), 1.E200);
    }
 
    // Second derivative : (-M^{-T}@I + Trace(M)*M^{-T}_{ik}M^{-T}_{lj}
    //                      -I@M^{-T} + Trace(M)*M^{-T}@M^{-T}/3)/(3det(M)^1/3)
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_matrix M(N, N);
      gmm::copy(args[0]->as_vector(), M.as_vector());
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      scalar_type det = bgeot::lu_inverse(&(*(M.begin())), N);
      if (det > 0) {
        base_tensor::iterator it = result.begin();
        for (size_type l = 0; l < N; ++l)
          for (size_type k = 0; k < N; ++k)
            for (size_type j = 0; j < N; ++j)
              for (size_type i = 0; i < N; ++i, ++it)
                *it = (- ((k == l) ? M(j, i) : scalar_type(0))
                       + tr*M(i,k)*M(l,j)
                       - ((i == j) ? M(l, k) : scalar_type(0))
                       + tr*M(j,i)*M(k,l)/ scalar_type(3))
                  / (scalar_type(3)*pow(det, scalar_type(1)/scalar_type(3)));
        GMM_ASSERT1(it == result.end(), "Internal error");
      } else
        std::fill(result.begin(), result.end(), 1.E200);
    }
  };
 
 
  // Matrix_j2 Operator
  struct matrix_j2_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_scalar_(sizes);
      return true;
    }
 
    // Value : i2(M)/(det(M)^2/3)
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_matrix M(N, N);
      gmm::copy(args[0]->as_vector(), M.as_vector());
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      scalar_type tr2 = scalar_type(0);
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j)
          tr2 += M(i,j)*M(j,i);
      scalar_type i2 = (tr*tr-tr2)/scalar_type(2);
      scalar_type det = bgeot::lu_det(&(*(M.begin())), N);
      if (det > 0)
        result[0] = i2 / pow(det, scalar_type(2)/scalar_type(3));
      else
        result[0] = 1.E200;
    }
 
    // Derivative : (Trace(M)*I-M^T-2i2(M)M^{-T}/3)/(det(M)^2/3)
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      const base_tensor &t = *args[0];
      base_matrix M(N, N);
      gmm::copy(t.as_vector(), M.as_vector());
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      scalar_type tr2 = scalar_type(0);
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j)
          tr2 += M(i,j)*M(j,i);
      scalar_type i2 = (tr*tr-tr2)/scalar_type(2);
      scalar_type det = bgeot::lu_inverse(&(*(M.begin())), N);
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < N; ++j)
        for (size_type i = 0; i < N; ++i, ++it)
          *it = (((i == j) ? tr : scalar_type(0)) - t[j+N*i]
                 - scalar_type(2)*i2*M(j,i)/(scalar_type(3)))
            / pow(det, scalar_type(2)/scalar_type(3));
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      const base_tensor &t = *args[0];
      base_matrix M(N, N);
      gmm::copy(t.as_vector(), M.as_vector());
      scalar_type tr = scalar_type(0);
      for (size_type i = 0; i < N; ++i) tr += M(i,i);
      scalar_type tr2 = scalar_type(0);
      for (size_type i = 0; i < N; ++i)
        for (size_type j = 0; j < N; ++j)
          tr2 += M(i,j)*M(j,i);
      scalar_type i2 = (tr*tr-tr2)/scalar_type(2);
      scalar_type det = bgeot::lu_inverse(&(*(M.begin())), N);
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < N; ++l)
        for (size_type k = 0; k < N; ++k)
          for (size_type j = 0; j < N; ++j)
            for (size_type i = 0; i < N; ++i, ++it)
              *it = ( + (((i==j) && (k==l)) ? 1. : 0.)
                      - (((i==l) && (k==j)) ? 1. : 0.)
                      + 10.*i2*M(j,i)*M(l,k)/(9.)
                      - 2.*(M(j,i)*(tr*((k==l) ? 1.:0.)-t[l+N*k]))/(3.)
                      - 2.*(M(l,k)*(tr*((i==j) ? 1.:0.)-t[j+N*i]))/(3.)
                      - 2.*i2*(M(j,i)*M(l,k)-M(j,k)*M(l,i))/(3.))
                / pow(det, scalar_type(2)/scalar_type(3));
    }
  };
 
  // Right-Cauchy-Green operator (F^{T}F)
  struct Right_Cauchy_Green_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[1]);
      return true;
    }
 
    // Value : F^{T}F
    void value(const arg_list &args, base_tensor &result) const {
      // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < n; ++j)
        for (size_type i = 0; i < n; ++i, ++it) {
          *it = scalar_type(0);
          for (size_type k = 0; k < m; ++k)
            *it += (*(args[0]))[i*m+k] *  (*(args[0]))[j*m+k];
        }
    }
 
    // Derivative : F{kj}delta{li}+F{ki}delta{lj}
    // (comes from H -> H^{T}F + F^{T}H)
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const { // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < n; ++l)
        for (size_type k = 0; k < m; ++k)
          for (size_type j = 0; j < n; ++j)
            for (size_type i = 0; i < n; ++i, ++it) {
              *it = scalar_type(0);
              if (l == i) *it += (*(args[0]))[j*m+k];
              if (l == j) *it += (*(args[0]))[i*m+k];
            }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative :
    // A{ijklop}=delta{ok}delta{li}delta{pj} + delta{ok}delta{pi}delta{lj}
    // comes from (H,K) -> H^{T}K + K^{T}H
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const { // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type p = 0; p < n; ++p)
        for (size_type o = 0; o < m; ++o)
          for (size_type l = 0; l < n; ++l)
            for (size_type k = 0; k < m; ++k)
              for (size_type j = 0; j < n; ++j)
                for (size_type i = 0; i < n; ++i, ++it) {
                  *it = scalar_type(0);
                  if (o == k) {
                    if (l == i && p == j) *it +=  scalar_type(1);
                    if (p == i && l == j) *it +=  scalar_type(1);
                  }
                }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
  };
 
  // Left-Cauchy-Green operator (FF^{T})
  struct Left_Cauchy_Green_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[0]);
      return true;
    }
 
    // Value : FF^{T}
    void value(const arg_list &args, base_tensor &result) const {
      // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < m; ++j)
        for (size_type i = 0; i < m; ++i, ++it) {
          *it = scalar_type(0);
          for (size_type k = 0; k < n; ++k)
            *it += (*(args[0]))[k*m+i] *  (*(args[0]))[k*m+j];
        }
    }
 
    // Derivative : F{jl}delta{ik}+F{il}delta{kj}
    // (comes from H -> HF^{T} + FH^{T})
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const {
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < n; ++l)
        for (size_type k = 0; k < m; ++k)
          for (size_type j = 0; j < m; ++j)
            for (size_type i = 0; i < m; ++i, ++it) {
              *it = scalar_type(0);
              if (k == i) *it += (*(args[0]))[l*m+j];
              if (k == j) *it += (*(args[0]))[l*m+i];
            }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative :
    // A{ijklop}=delta{ki}delta{lp}delta{oj} + delta{oi}delta{pl}delta{kj}
    // comes from (H,K) -> HK^{T} + KH^{T}
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const {
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type p = 0; p < n; ++p)
        for (size_type o = 0; o < m; ++o)
          for (size_type l = 0; l < n; ++l)
            for (size_type k = 0; k < m; ++k)
              for (size_type j = 0; j < m; ++j)
                for (size_type i = 0; i < m; ++i, ++it) {
                  *it = scalar_type(0);
                  if (p == l) {
                    if (k == i && o == j) *it +=  scalar_type(1);
                    if (o == i && k == j) *it +=  scalar_type(1);
                  }
                }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
  };
 
 
  // Green-Lagrangian operator (F^{T}F-I)/2
  struct Green_Lagrangian_operator : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 1 || args[0]->sizes().size() != 2) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[1]);
      return true;
    }
 
    // Value : F^{T}F
    void value(const arg_list &args, base_tensor &result) const {
      // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < n; ++j)
        for (size_type i = 0; i < n; ++i, ++it) {
          *it = scalar_type(0);
          for (size_type k = 0; k < m; ++k)
            *it += (*(args[0]))[i*m+k]*(*(args[0]))[j*m+k]*scalar_type(0.5);
          if (i == j) *it -= scalar_type(0.5);
        }
    }
 
    // Derivative : (F{kj}delta{li}+F{ki}delta{lj})/2
    // (comes from H -> (H^{T}F + F^{T}H)/2)
    void derivative(const arg_list &args, size_type,
                    base_tensor &result) const { // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < n; ++l)
        for (size_type k = 0; k < m; ++k)
          for (size_type j = 0; j < n; ++j)
            for (size_type i = 0; i < n; ++i, ++it) {
              *it = scalar_type(0);
              if (l == i) *it += (*(args[0]))[j*m+k]*scalar_type(0.5);
              if (l == j) *it += (*(args[0]))[i*m+k]*scalar_type(0.5);
            }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative :
    // A{ijklop}=(delta{ok}delta{li}delta{pj} + delta{ok}delta{pi}delta{lj})/2
    // comes from (H,K) -> (H^{T}K + K^{T}H)/2
    void second_derivative(const arg_list &args, size_type, size_type,
                           base_tensor &result) const { // to be verified
      size_type m = args[0]->sizes()[0], n = args[0]->sizes()[1];
      base_tensor::iterator it = result.begin();
      for (size_type p = 0; p < n; ++p)
        for (size_type o = 0; o < m; ++o)
          for (size_type l = 0; l < n; ++l)
            for (size_type k = 0; k < m; ++k)
              for (size_type j = 0; j < n; ++j)
                for (size_type i = 0; i < n; ++i, ++it) {
                  *it = scalar_type(0);
                  if (o == k) {
                    if (l == i && p == j) *it +=  scalar_type(0.5);
                    if (p == i && l == j) *it +=  scalar_type(0.5);
                  }
                }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
  };
 
  // Cauchy stress tensor from second Piola-Kirchhoff stress tensor
  // (I+Grad_u)\sigma(I+Grad_u')/det(I+Grad_u)
  struct Cauchy_stress_from_PK2 : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 2 || args[0]->sizes().size() != 2
          || args[1]->sizes().size() != 2
          || args[0]->sizes()[0] !=  args[0]->sizes()[1]
          || args[1]->sizes()[0] !=  args[0]->sizes()[1]
          || args[1]->sizes()[1] !=  args[0]->sizes()[1]) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[1]);
      return true;
    }
 
    // Value : (I+Grad_u)\sigma(I+Grad_u')/det(I+Grad_u)
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_matrix F(N, N), sigma(N,N), aux(N, N);
      gmm::copy(args[0]->as_vector(), sigma.as_vector());
      gmm::copy(args[1]->as_vector(), F.as_vector());
      gmm::add(gmm::identity_matrix(), F);
      gmm::mult(F, sigma, aux);
      gmm::mult(aux, gmm::transposed(F), sigma);
      scalar_type det = bgeot::lu_det(&(*(F.begin())), N);
      gmm::scale(sigma, scalar_type(1)/det);
      gmm::copy(sigma.as_vector(), result.as_vector());
    }
 
    // Derivative :
    void derivative(const arg_list &args, size_type nder,
                    base_tensor &result) const { // to be verified
      size_type N = args[0]->sizes()[0];
      base_matrix F(N, N);
      gmm::copy(args[1]->as_vector(), F.as_vector());
      gmm::add(gmm::identity_matrix(), F);
      scalar_type det = bgeot::lu_det(&(*(F.begin())), N);
 
      base_tensor::iterator it = result.begin();
 
      switch (nder) {
      case 1: // Derivative with respect to sigma : F_ik F_jl / det(F)
        for (size_type l = 0; l < N; ++l)
          for (size_type k = 0; k < N; ++k)
            for (size_type j = 0; j < N; ++j)
              for (size_type i = 0; i < N; ++i, ++it)
                *it = F(i,k) * F(j,l) / det;
        break;
 
      case 2:
        {
          // Derivative with respect to Grad_u :
          // (delta_ik sigma_lm F_jm + F_im sigma_mk delta_lj
          //   - F_im sigma_mn F_jn F^{-1}_lk) / det(F)
          base_matrix sigma(N,N), aux(N,N), aux2(N,N);
          gmm::copy(args[0]->as_vector(), sigma.as_vector());
          gmm::mult(sigma, gmm::transposed(F), aux);
          gmm::mult(F, aux, aux2);
          bgeot::lu_inverse(&(*(F.begin())), N);
          for (size_type l = 0; l < N; ++l)
            for (size_type k = 0; k < N; ++k)
              for (size_type j = 0; j < N; ++j)
                for (size_type i = 0; i < N; ++i, ++it) {
                  *it = scalar_type(0);
                  if (i == k) *it += aux(l, j) / det;
                  if (l == j) *it += aux(k, i) / det;
                  *it -= aux2(i,j) * F(l,k) / det;
                }
        }
        break;
      }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative : to be implemented
    void second_derivative(const arg_list &, size_type, size_type,
                           base_tensor &) const {
      GMM_ASSERT1(false, "Sorry, not implemented");
    }
  };
 
 
  struct AHL_wrapper_sigma : public ga_nonlinear_operator {
    phyperelastic_law AHL;
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 2 || args[0]->sizes().size() != 2
          || args[1]->size() != AHL->nb_params()
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[0]);
      return true;
    }
 
    // Value :
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_vector params(AHL->nb_params());
      gmm::copy(args[1]->as_vector(), params);
      base_matrix Gu(N, N), E(N,N), sigma(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      gmm::add(gmm::identity_matrix(), Gu);
      scalar_type det = bgeot::lu_det(&(*(Gu.begin())), N);
 
      AHL->sigma(E, sigma, params, det);
      gmm::copy(sigma.as_vector(), result.as_vector());
    }
 
    // Derivative :
    void derivative(const arg_list &args, size_type nder,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_vector params(AHL->nb_params());
      gmm::copy(args[1]->as_vector(), params);
      base_tensor grad_sigma(N, N, N, N);
      base_matrix Gu(N, N), E(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      gmm::add(gmm::identity_matrix(), Gu);
      scalar_type det = bgeot::lu_det(&(*(Gu.begin())), N);
 
      GMM_ASSERT1(nder == 1, "Sorry, the derivative of this hyperelastic "
                  "law with respect to its parameters is not available.");
 
      AHL->grad_sigma(E, grad_sigma, params, det);
 
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < N; ++l)
        for (size_type k = 0; k < N; ++k)
          for (size_type j = 0; j < N; ++j)
            for (size_type i = 0; i < N; ++i, ++it) {
              *it = scalar_type(0);
              for (size_type m = 0; m < N; ++m)
                *it += grad_sigma(i,j,m,l) * Gu(k, m); // to be optimized
            }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
 
    // Second derivative : not implemented (not necessary)
    void second_derivative(const arg_list &, size_type, size_type,
                           base_tensor &) const {
      GMM_ASSERT1(false, "Sorry, second derivative not implemented");
    }
 
    AHL_wrapper_sigma(const phyperelastic_law &A) : AHL(A) {}
 
  };
 
 
  struct AHL_wrapper_potential : public ga_nonlinear_operator {
    phyperelastic_law AHL;
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 2 || args[0]->sizes().size() != 2
          || args[1]->size() != AHL->nb_params()
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_scalar_(sizes);
      return true;
    }
 
    // Value :
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_vector params(AHL->nb_params());
      gmm::copy(args[1]->as_vector(), params);
      base_matrix Gu(N, N), E(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      gmm::add(gmm::identity_matrix(), Gu);
      scalar_type det = bgeot::lu_det(&(*(Gu.begin())), N); // not necessary here
 
      result[0] = AHL->strain_energy(E, params, det);
    }
 
    // Derivative :
    void derivative(const arg_list &args, size_type nder,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      base_vector params(AHL->nb_params());
      gmm::copy(args[1]->as_vector(), params);
      base_matrix Gu(N, N), E(N,N), sigma(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      gmm::add(gmm::identity_matrix(), Gu);
      scalar_type det = bgeot::lu_det(&(*(Gu.begin())), N); // not necessary here
 
      GMM_ASSERT1(nder == 1, "Sorry, Cannot derive the potential with "
                  "respect to law parameters.");
 
      AHL->sigma(E, sigma, params, det);
      gmm::mult(Gu, sigma, E);
      gmm::copy(E.as_vector(), result.as_vector());
    }
 
 
    // Second derivative :
    void second_derivative(const arg_list &args, size_type nder1,
                           size_type nder2, base_tensor &result) const {
 
      size_type N = args[0]->sizes()[0];
      base_vector params(AHL->nb_params());
      gmm::copy(args[1]->as_vector(), params);
      base_tensor grad_sigma(N, N, N, N);
      base_matrix Gu(N, N), E(N,N), sigma(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      gmm::add(gmm::identity_matrix(), Gu);
      scalar_type det = bgeot::lu_det(&(*(Gu.begin())), N);
 
      GMM_ASSERT1(nder1 == 1 && nder2 == 1, "Sorry, Cannot derive the "
                  "potential with respect to law parameters.");
 
      AHL->sigma(E, sigma, params, det);
      AHL->grad_sigma(E, grad_sigma, params, det);
 
      base_tensor::iterator it = result.begin();
      for (size_type l = 0; l < N; ++l)
        for (size_type k = 0; k < N; ++k)
          for (size_type j = 0; j < N; ++j)
            for (size_type i = 0; i < N; ++i, ++it) {
              *it = scalar_type(0);
              if (i == k) *it += sigma(l,j);
 
              for (size_type m = 0; m < N; ++m) // to be optimized
                for (size_type n = 0; n < N; ++n)
                  *it += grad_sigma(n,j,m,l) * Gu(k, m) * Gu(i, n);
            }
      GMM_ASSERT1(it == result.end(), "Internal error");
 
    }
 
    AHL_wrapper_potential(const phyperelastic_law &A) : AHL(A) {}
 
  };
 
 
  // Saint-Venant Kirchhoff tensor:
  // lambda Tr(E)Id + 2*mu*E with E = (Grad_u+Grad_u'+Grad_u'Grad_u)/2
  struct Saint_Venant_Kirchhoff_sigma : public ga_nonlinear_operator {
    bool result_size(const arg_list &args, bgeot::multi_index &sizes) const {
      if (args.size() != 2 || args[0]->sizes().size() != 2
          || args[1]->size() != 2
          || args[0]->sizes()[0] != args[0]->sizes()[1]) return false;
      ga_init_square_matrix_(sizes, args[0]->sizes()[0]);
      return true;
    }
 
    // Value : Tr(E) + 2*mu*E
    void value(const arg_list &args, base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      scalar_type lambda = (*(args[1]))[0], mu = (*(args[1]))[1];
      base_matrix Gu(N, N), E(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      gmm::mult(gmm::transposed(Gu), Gu, E);
      gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
      gmm::scale(E, scalar_type(0.5));
      scalar_type trE = gmm::mat_trace(E);
 
      base_tensor::iterator it = result.begin();
      for (size_type j = 0; j < N; ++j)
        for (size_type i = 0; i < N; ++i, ++it) {
          *it = 2*mu*E(i,j);
          if (i == j) *it += lambda*trE;
        }
    }
 
    // Derivative / u: lambda Tr(H + H'*U) Id + mu(H + H' + H'*U + U'*H)
    //                 with U = Grad_u, H = Grad_Test_u
    // Implementation: A{ijkl} = lambda(delta{kl}delta{ij} + U{kl}delta{ij})
    //        + mu(delta{ik}delta{jl} + delta{il}delta{jk} + U{kj}delta{il} +
    //             U{ki}delta{lj})
    void derivative(const arg_list &args, size_type nder,
                    base_tensor &result) const {
      size_type N = args[0]->sizes()[0];
      scalar_type lambda = (*(args[1]))[0], mu = (*(args[1]))[1], trE;
      base_matrix Gu(N, N), E(N,N);
      gmm::copy(args[0]->as_vector(), Gu.as_vector());
      if (nder > 1) {
        gmm::mult(gmm::transposed(Gu), Gu, E);
        gmm::add(Gu, E); gmm::add(gmm::transposed(Gu), E);
        gmm::scale(E, scalar_type(0.5));
      }
      base_tensor::iterator it = result.begin();
      switch (nder) {
      case 1: // Derivative with respect to u
        for (size_type l = 0; l < N; ++l)
          for (size_type k = 0; k < N; ++k)
            for (size_type j = 0; j < N; ++j)
              for (size_type i = 0; i < N; ++i, ++it) {
                *it = scalar_type(0);
                if (i == j && k == l) *it += lambda;
                if (i == j) *it += lambda*Gu(k,l);
                if (i == k && j == l) *it += mu;
                if (i == l && j == k) *it += mu;
                if (i == l) *it += mu*Gu(k,j);
                if (l == j) *it += mu*Gu(k,i);
              }
        break;
      case 2:
        // Derivative with respect to lambda: Tr(E)Id
        trE = gmm::mat_trace(E);
        for (size_type j = 0; j < N; ++j)
          for (size_type i = 0; i < N; ++i, ++it) {
            *it = scalar_type(0); if (i == j) *it += trE;
          }
        // Derivative with respect to mu: 2E
        for (size_type j = 0; j < N; ++j)
          for (size_type i = 0; i < N; ++i, ++it) {
            *it += 2*E(i,j);
          }
        break;
      default: GMM_ASSERT1(false, "Internal error");
      }
      GMM_ASSERT1(it == result.end(), "Internal error");
    }
 
    // Second derivative : not implemented
    void second_derivative(const arg_list &, size_type, size_type,
                           base_tensor &) const {
      GMM_ASSERT1(false, "Sorry, second derivative not implemented");
    }
  };
 
 
 
  static bool init_predef_operators() {
 
    ga_predef_operator_tab &PREDEF_OPERATORS
      = dal::singleton<ga_predef_operator_tab>::instance();
 
    PREDEF_OPERATORS.add_method
      ("Matrix_i2", std::make_shared<matrix_i2_operator>());
    PREDEF_OPERATORS.add_method
      ("Matrix_j1", std::make_shared<matrix_j1_operator>());
    PREDEF_OPERATORS.add_method
      ("Matrix_j2", std::make_shared<matrix_j2_operator>());
    PREDEF_OPERATORS.add_method
      ("Right_Cauchy_Green", std::make_shared<Right_Cauchy_Green_operator>());
    PREDEF_OPERATORS.add_method
      ("Left_Cauchy_Green", std::make_shared<Left_Cauchy_Green_operator>());
    PREDEF_OPERATORS.add_method
      ("Green_Lagrangian", std::make_shared<Green_Lagrangian_operator>());
 
    PREDEF_OPERATORS.add_method
      ("Cauchy_stress_from_PK2", std::make_shared<Cauchy_stress_from_PK2>());
 
    PREDEF_OPERATORS.add_method
      ("Saint_Venant_Kirchhoff_sigma",
       std::make_shared<Saint_Venant_Kirchhoff_sigma>());
    PREDEF_OPERATORS.add_method
      ("Saint_Venant_Kirchhoff_PK2",
       std::make_shared<Saint_Venant_Kirchhoff_sigma>());
    PREDEF_OPERATORS.add_method
      ("Saint_Venant_Kirchhoff_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<SaintVenant_Kirchhoff_hyperelastic_law>()));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Saint_Venant_Kirchhoff_sigma",
       std::make_shared<Saint_Venant_Kirchhoff_sigma>());
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Saint_Venant_Kirchhoff_PK2",
       std::make_shared<Saint_Venant_Kirchhoff_sigma>());
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Saint_Venant_Kirchhoff_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<SaintVenant_Kirchhoff_hyperelastic_law>()));
 
    phyperelastic_law gbklaw
      = std::make_shared<generalized_Blatz_Ko_hyperelastic_law>();
    PREDEF_OPERATORS.add_method
      ("Generalized_Blatz_Ko_sigma",
       std::make_shared<AHL_wrapper_sigma>(gbklaw));
     PREDEF_OPERATORS.add_method
      ("Generalized_Blatz_Ko_PK2",
       std::make_shared<AHL_wrapper_sigma>(gbklaw));
   PREDEF_OPERATORS.add_method
      ("Generalized_Blatz_Ko_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<generalized_Blatz_Ko_hyperelastic_law>()));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Generalized_Blatz_Ko_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(gbklaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Generalized_Blatz_Ko_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(gbklaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Generalized_Blatz_Ko_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(gbklaw)));
 
    phyperelastic_law cigelaw
      = std::make_shared<Ciarlet_Geymonat_hyperelastic_law>();
    PREDEF_OPERATORS.add_method
      ("Ciarlet_Geymonat_PK2", std::make_shared<AHL_wrapper_sigma>(cigelaw));
    PREDEF_OPERATORS.add_method
      ("Ciarlet_Geymonat_sigma", std::make_shared<AHL_wrapper_sigma>(cigelaw));
    PREDEF_OPERATORS.add_method
      ("Ciarlet_Geymonat_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Ciarlet_Geymonat_hyperelastic_law>()));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Ciarlet_Geymonat_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cigelaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Ciarlet_Geymonat_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cigelaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Ciarlet_Geymonat_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(cigelaw)));
 
    phyperelastic_law morilaw
      = std::make_shared<Mooney_Rivlin_hyperelastic_law>();
    PREDEF_OPERATORS.add_method
      ("Incompressible_Mooney_Rivlin_sigma",
       std::make_shared<AHL_wrapper_sigma>(morilaw));
    PREDEF_OPERATORS.add_method
      ("Incompressible_Mooney_Rivlin_PK2",
       std::make_shared<AHL_wrapper_sigma>(morilaw));
    PREDEF_OPERATORS.add_method
      ("Incompressible_Mooney_Rivlin_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Mooney_Rivlin_hyperelastic_law>()));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Mooney_Rivlin_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(morilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Mooney_Rivlin_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(morilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Mooney_Rivlin_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(morilaw)));
 
    phyperelastic_law cmorilaw
      = std::make_shared<Mooney_Rivlin_hyperelastic_law>(true);
    PREDEF_OPERATORS.add_method
      ("Compressible_Mooney_Rivlin_sigma",
       std::make_shared<AHL_wrapper_sigma>(cmorilaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Mooney_Rivlin_PK2",
       std::make_shared<AHL_wrapper_sigma>(cmorilaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Mooney_Rivlin_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Mooney_Rivlin_hyperelastic_law>(true)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Mooney_Rivlin_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cmorilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Mooney_Rivlin_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cmorilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Mooney_Rivlin_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(cmorilaw)));
 
    phyperelastic_law ineolaw
      = std::make_shared<Mooney_Rivlin_hyperelastic_law>(false, true);
    PREDEF_OPERATORS.add_method
      ("Incompressible_Neo_Hookean_sigma",
       std::make_shared<AHL_wrapper_sigma>(ineolaw));
    PREDEF_OPERATORS.add_method
      ("Incompressible_Neo_Hookean_PK2",
       std::make_shared<AHL_wrapper_sigma>(ineolaw));
    PREDEF_OPERATORS.add_method
      ("Incompressible_Neo_Hookean_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Mooney_Rivlin_hyperelastic_law>(false,true)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Neo_Hookean_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(ineolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Neo_Hookean_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(ineolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Incompressible_Neo_Hookean_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(ineolaw)));
 
    phyperelastic_law cneolaw
      = std::make_shared<Mooney_Rivlin_hyperelastic_law>(true, true);
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_sigma",
       std::make_shared<AHL_wrapper_sigma>(cneolaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_PK2",
       std::make_shared<AHL_wrapper_sigma>(cneolaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Mooney_Rivlin_hyperelastic_law>(true,true)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(cneolaw)));
 
    phyperelastic_law cneobolaw
      = std::make_shared<Neo_Hookean_hyperelastic_law>(true);
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Bonet_sigma",
       std::make_shared<AHL_wrapper_sigma>(cneobolaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Bonet_PK2",
       std::make_shared<AHL_wrapper_sigma>(cneobolaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Bonet_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Neo_Hookean_hyperelastic_law>(true)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Bonet_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneobolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Bonet_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneobolaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Bonet_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(cneobolaw)));
 
    phyperelastic_law cneocilaw
      = std::make_shared<Neo_Hookean_hyperelastic_law>(false);
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Ciarlet_sigma",
       std::make_shared<AHL_wrapper_sigma>(cneocilaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Ciarlet_PK2",
       std::make_shared<AHL_wrapper_sigma>(cneocilaw));
    PREDEF_OPERATORS.add_method
      ("Compressible_Neo_Hookean_Ciarlet_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<Neo_Hookean_hyperelastic_law>(false)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Ciarlet_sigma",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneocilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Ciarlet_PK2",
       std::make_shared<AHL_wrapper_sigma>
       (std::make_shared<plane_strain_hyperelastic_law>(cneocilaw)));
    PREDEF_OPERATORS.add_method
      ("Plane_Strain_Compressible_Neo_Hookean_Ciarlet_potential",
       std::make_shared<AHL_wrapper_potential>
       (std::make_shared<plane_strain_hyperelastic_law>(cneocilaw)));
 
    return true;
  }
 
  // declared in getfem_generic_assembly.cc
  bool predef_operators_nonlinear_elasticity_initialized
    = init_predef_operators();
 
 
  std::string adapt_law_name(const std::string &lawname, size_type N) {
    std::string adapted_lawname = lawname;
 
    for (size_type i = 0; i < lawname.size(); ++i)
      if (adapted_lawname[i] == ' ') adapted_lawname[i] = '_';
 
    if (adapted_lawname.compare("SaintVenant_Kirchhoff") == 0) {
      adapted_lawname = "Saint_Venant_Kirchhoff";
    } else if (adapted_lawname.compare("Saint_Venant_Kirchhoff") == 0) {
 
    } else if (adapted_lawname.compare("Generalized_Blatz_Ko") == 0) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else if (adapted_lawname.compare("Ciarlet_Geymonat") == 0) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else if (adapted_lawname.compare("Incompressible_Mooney_Rivlin") == 0) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else if (adapted_lawname.compare("Compressible_Mooney_Rivlin") == 0) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else if (adapted_lawname.compare("Incompressible_Neo_Hookean") == 0) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else if (adapted_lawname.compare("Compressible_Neo_Hookean") == 0 ||
               adapted_lawname.compare("Compressible_Neo_Hookean_Bonet") == 0 ||
               adapted_lawname.compare("Compressible_Neo_Hookean_Ciarlet") == 0 ) {
      if (N == 2) adapted_lawname = "Plane_Strain_" + adapted_lawname;
    } else
      GMM_ASSERT1(false, lawname << " is not a known hyperelastic law");
 
    return adapted_lawname;
  }
 
 
  size_type add_finite_strain_elasticity_brick
  (model &md, const mesh_im &mim, const std::string &lawname,
   const std::string &varname, const std::string &params,
   size_type region) {
    std::string test_varname = "Test_" + sup_previous_and_dot_to_varname(varname);
    size_type N = mim.linked_mesh().dim();
    GMM_ASSERT1(N >= 2 && N <= 3,
                "Finite strain elasticity brick works only in 2D or 3D");
 
    const mesh_fem *mf = md.pmesh_fem_of_variable(varname);
    GMM_ASSERT1(mf, "Finite strain elasticity brick can only be applied on "
                "fem variables");
    size_type Q = mf->get_qdim();
    GMM_ASSERT1(Q == N, "Finite strain elasticity brick can only be applied "
                "on a fem variable having the same dimension as the mesh");
 
    std::string adapted_lawname = adapt_law_name(lawname, N);
 
    std::string expr = "((Id(meshdim)+Grad_"+varname+")*(" + adapted_lawname
      + "_PK2(Grad_"+varname+","+params+"))):Grad_" + test_varname;
 
    return add_nonlinear_term
      (md, mim, expr, region, true, false,
       "Finite strain elasticity brick for " + adapted_lawname + " law");
  }
 
  size_type add_finite_strain_incompressibility_brick
  (model &md, const mesh_im &mim, const std::string &varname,
   const std::string &multname, size_type region) {
    std::string test_varname = "Test_" + sup_previous_and_dot_to_varname(varname);
    std::string test_multname = "Test_" + sup_previous_and_dot_to_varname(multname);
 
    std::string expr
      = "(" + test_multname+ ")*(1-Det(Id(meshdim)+Grad_" + varname + "))"
      + "-(" + multname + ")*(Det(Id(meshdim)+Grad_" + varname + ")"
      + "*((Inv(Id(meshdim)+Grad_" + varname + "))':Grad_"
      + test_varname + "))" ;
    return add_nonlinear_term
      (md, mim, expr, region, true, false,
       "Finite strain incompressibility brick");
  }
 
  void compute_finite_strain_elasticity_Von_Mises
    (model &md, const std::string &lawname, const std::string &varname,
     const std::string &params, const mesh_fem &mf_vm,
     model_real_plain_vector &VM, const mesh_region &rg) {
    size_type N = mf_vm.linked_mesh().dim();
 
    std::string adapted_lawname = adapt_law_name(lawname, N);
 
    std::string expr = "sqrt(3/2)*Norm(Deviator(Cauchy_stress_from_PK2("
      + adapted_lawname + "_PK2(Grad_" + varname + "," + params + "),Grad_"
      + varname + ")))";
    ga_interpolation_Lagrange_fem(md, expr, mf_vm, VM, rg);
  }
 
 
 
}  /* end of namespace getfem.                                             */