easfunctions/displacementgradient.hh

Go to the documentation of this file.

// SPDX-FileCopyrightText: 2021-2025 The Ikarus Developers mueller@ibb.uni-stuttgart.de
// SPDX-License-Identifier: LGPL-3.0-or-later
 
#pragma once
 
#include <dune/common/fvector.hh>
#include <dune/localfefunctions/eigenDuneTransformations.hh>
#include <dune/localfefunctions/expressions/greenLagrangeStrains.hh>
 
#include <Eigen/Core>
 
#include <ikarus/utils/tensorutils.hh>
 
namespace Ikarus::EAS {
struct DisplacementGradient
{
  template <typename GEO, typename EAST>
  static auto value(const GEO& geo, const auto& uFunction, const Dune::FieldVector<double, GEO::mydimension>& gpPos,
                    const EAST& easFunction, const auto& alpha) {
    using ST                                = typename std::remove_cvref_t<decltype(uFunction)>::ctype;
    constexpr int myDim                     = GEO::mydimension;
    const Eigen::Matrix<ST, myDim, myDim> H = computeDisplacementGradient(geo, uFunction, gpPos, easFunction, alpha);
    const Eigen::Matrix<ST, myDim, myDim> E = 0.5 * (H + H.transpose() + H.transpose() * H);
    return toVoigt(E, true).eval();
  }
 
  template <int wrtCoeff, typename GEO, typename EAST>
  static auto firstDerivative(const GEO& geo, const auto& uFunction, const auto& localBasis, const auto& gpIndex,
                              const Dune::FieldVector<double, GEO::mydimension>& gpPos, const EAST& easFunction,
                              const auto& alpha, const int node = sNaN) {
    using namespace Dune::DerivativeDirections;
    using namespace Dune;
    using ST            = typename std::remove_cvref_t<decltype(uFunction)>::ctype;
    constexpr int myDim = GEO::mydimension;
    static_assert(myDim == 2 or myDim == 3,
                  "An enhancement of displacement gradient can only be performed for the 2D or 3D case.");
    constexpr int strainSize = myDim * (myDim + 1) / 2;
    using MatrixType         = Eigen::Matrix<ST, myDim, myDim>;
 
    const MatrixType F = computeDeformationGradient(geo, uFunction, gpPos, easFunction, alpha)
                             .transpose()
                             .eval(); // transposed such that the rows are u_{,1} and u_{,2}
    const auto g0 = F.row(0);
    const auto g1 = F.row(1);
 
    if constexpr (wrtCoeff == 0) { // E,d
      const auto gradUdI = uFunction.evaluateDerivative(gpIndex, wrt(spatialAll, coeff(node)), on(gridElement));
 
      Eigen::Vector<ST, myDim> dN;
      for (const auto i : Dune::range(myDim))
        dN[i] = getDiagonalEntry(gradUdI[i], 0);
      Eigen::Matrix<ST, strainSize, myDim> bopI;
      bopI.row(0) = dN[0] * g0;
      bopI.row(1) = dN[1] * g1;
      if constexpr (myDim == 2)
        bopI.row(2) = dN[1] * g0 + dN[0] * g1;
      else {
        const auto g2 = F.row(2);
        bopI.row(2)   = dN[2] * g2;
        bopI.row(3)   = dN[2] * g1 + dN[1] * g2;
        bopI.row(4)   = dN[2] * g0 + dN[0] * g2;
        bopI.row(5)   = dN[1] * g0 + dN[0] * g1;
      }
      return bopI.eval();
    } else if constexpr (wrtCoeff == 1) { // E,a
      const typename EAST::HType Harray = easFunction(gpPos);
      constexpr int enhancedStrainSize  = EAST::enhancedStrainSize;
      Eigen::Matrix<ST, strainSize, enhancedStrainSize> M;
      M.setZero(); // zeros returned if Harray is empty
      for (const auto p : Dune::range(enhancedStrainSize)) {
        const MatrixType Htilde = Harray[p];
        M(0, p)                 = Htilde.col(0).dot(g0);
        M(1, p)                 = Htilde.col(1).dot(g1);
        if constexpr (myDim == 2)
          M(2, p) = Htilde.col(1).dot(g0) + Htilde.col(0).dot(g1);
        else {
          const auto g2 = F.row(2);
          M(2, p)       = Htilde.col(2).dot(g2);
          M(3, p)       = Htilde.col(2).dot(g1) + Htilde.col(1).dot(g2);
          M(4, p)       = Htilde.col(2).dot(g0) + Htilde.col(0).dot(g2);
          M(5, p)       = Htilde.col(1).dot(g0) + Htilde.col(0).dot(g1);
        }
      }
      return M.eval();
    } else
      static_assert(Dune::AlwaysFalse<GEO>::value,
                    "firstDerivative can only be called with wrtCoeff as 0 and 1 indicating first derivative of the "
                    "Green-Lagrange strain w.r.t d and "
                    "alpha, respectively.");
  }
 
  template <int wrtCoeff, typename GEO, typename ST, typename EAST>
  static auto secondDerivative(const GEO& geo, const auto& uFunction, const auto& localBasis, const auto& gpIndex,
                               const Dune::FieldVector<double, GEO::mydimension>& gpPos,
                               const Eigen::Vector<ST, GEO::mydimension*(GEO::mydimension + 1) / 2>& S,
                               const EAST& easFunction, const auto& alpha, const int I = sNaN, const int J = sNaN) {
    constexpr int myDim = GEO::mydimension;
    using namespace Dune::DerivativeDirections;
    using namespace Dune;
    static_assert(myDim == 2 or myDim == 3,
                  "An enhancement of displacement gradient can only be performed for the 2D or 3D case.");
    if constexpr (wrtCoeff == 0) { // E_dd = Ec_dd (as Etilde,dd = 0)
      const auto strainFunction = Dune::greenLagrangeStrains(uFunction);
      const auto kg = strainFunction.evaluateDerivative(gpIndex, wrt(coeff(I, J)), along(S), on(gridElement));
      return kg.eval();
    } else if constexpr (wrtCoeff == 1) { // E_aa
      constexpr int enhancedStrainSize  = EAST::enhancedStrainSize;
      const typename EAST::HType Harray = easFunction(gpPos);
      auto kg                           = Eigen::Matrix<ST, enhancedStrainSize, enhancedStrainSize>::Zero().eval();
 
      for (const auto i : Dune::range(myDim)) {
        for (const auto j : Dune::range(myDim)) {
          const auto sij = 0.5 * S[toVoigt<myDim>(i, j)];
          for (const auto p : Dune::range(enhancedStrainSize))
            for (const auto q : Dune::range(enhancedStrainSize)) {
              const auto Hq = Harray[q];
              kg(p, q) += sij * (Harray[p].col(i).transpose() * Harray[q].col(j) +
                                 Harray[p].col(j).transpose() * Harray[q].col(i))
                                    .sum();
            }
        }
      }
      return kg;
    } else if constexpr (wrtCoeff == 2) { // E_ad
      constexpr int enhancedStrainSize  = EAST::enhancedStrainSize;
      const typename EAST::HType Harray = easFunction(gpPos);
      auto kg                           = Eigen::Matrix<ST, enhancedStrainSize, myDim>::Zero().eval();
      const auto gradUdI = uFunction.evaluateDerivative(gpIndex, wrt(spatialAll, coeff(I)), on(gridElement));
 
      std::array<Eigen::Matrix<ST, myDim, myDim>, myDim> dNIdT;
      std::ranges::fill(dNIdT, Eigen::Matrix<ST, myDim, myDim>::Zero());
      for (const auto i : Dune::range(myDim))
        for (const auto j : Dune::range(myDim))
          dNIdT[i](i, j) = getDiagonalEntry(gradUdI[j], 0);
 
      for (const auto i : Dune::range(myDim)) {
        for (const auto j : Dune::range(myDim)) {
          const auto sij = 0.5 * S[toVoigt<myDim>(i, j)];
          for (const auto p : Dune::range(enhancedStrainSize))
            for (const auto q : Dune::range(myDim)) {
              kg(p, q) += sij * (Harray[p].col(i).transpose() * dNIdT[q].col(j) +
                                 Harray[p].col(j).transpose() * dNIdT[q].col(i))
                                    .sum();
            }
        }
      }
      return kg;
    } else
      static_assert(
          Dune::AlwaysFalse<GEO>::value,
          "secondDerivative can only be called with wrtCoeff as 0, 1 and 2 indicating second derivative of the "
          "Green-Lagrange strain w.r.t d and "
          "alpha and the mixed derivative, respectively.");
  }
 
  static constexpr auto name() { return std::string("Displacement Gradient"); }
 
private:
  static constexpr int sNaN = std::numeric_limits<int>::signaling_NaN();
 
  template <typename GEO, typename EAST>
  static auto computeDisplacementGradient(const GEO& geo, const auto& uFunction,
                                          const Dune::FieldVector<double, GEO::mydimension>& gpPos,
                                          const EAST& easFunction, const auto& alpha) {
    using ST            = typename std::remove_cvref_t<decltype(uFunction)>::ctype;
    constexpr int myDim = GEO::mydimension;
    using MatrixType    = Eigen::Matrix<ST, myDim, myDim>;
 
    using namespace Dune::DerivativeDirections;
    using namespace Dune;
 
    const MatrixType Hc = toEigen(uFunction.evaluateDerivative(gpPos, wrt(spatialAll), on(gridElement))).eval();
    const typename EAST::HType Harray = easFunction(gpPos);
    constexpr int enhancedStrainSize  = EAST::enhancedStrainSize;
    typename EAST::AnsatzType Htilde;
    Htilde.setZero(); // zeros returned if Harray is empty
    for (const auto p : Dune::range(enhancedStrainSize))
      Htilde += Harray[p] * alpha[p];
    const MatrixType H = Hc + Htilde;
    return H;
  }
 
  template <typename GEO, typename EAST>
  static auto computeDeformationGradient(const GEO& geo, const auto& uFunction,
                                         const Dune::FieldVector<double, GEO::mydimension>& gpPos,
                                         const EAST& easFunction, const auto& alpha) {
    using ST            = typename std::remove_cvref_t<decltype(uFunction)>::ctype;
    constexpr int myDim = GEO::mydimension;
    using MatrixType    = Eigen::Matrix<ST, myDim, myDim>;
    const MatrixType H  = computeDisplacementGradient(geo, uFunction, gpPos, easFunction, alpha);
    const MatrixType F  = (MatrixType::Identity() + H).eval();
    return F;
  }
};
 
} // namespace Ikarus::EAS