example.hpp

Go to the documentation of this file.

// Copyright (c) 2010-2024, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
 
#ifndef ADEXAMPLE_HPP
#define ADEXAMPLE_HPP
 
#include "mfem.hpp"
#include "admfem.hpp"
#include <memory>
#include <iostream>
#include <fstream>
 
namespace mfem
{
 
/// Example: Implementation of the residual evaluation for p-Laplacian
/// problem. The residual is evaluated at the integration points for PDE
/// parameters vparam and state fields (derivatives with respect to x,y,z and
/// primal field) stored in vector uu.
template<typename TDataType, typename TParamVector, typename TStateVector,
         int residual_size, int state_size, int param_size>
class MyResidualFunctor
{
public:
   /// The operator returns the first derivative of the energy with respect to
   /// all state variables. These are set in vector uu and consist of the
   /// derivatives with respect to x,y,z and the primal field. The derivative is
   /// stored in vector rr with length equal to the length of vector uu.
   void operator()(TParamVector &vparam, TStateVector &uu, TStateVector &rr)
   {
      MFEM_ASSERT(residual_size==4,
                  "PLaplacianResidual residual_size should be equal to 4!");
      real_t pp = vparam[0];
      real_t ee = vparam[1];
      real_t ff = vparam[2];
 
      // The vector rr holds the gradients of the following expression:
      // (u_x^2+u_y^2+u_z^2+\varepsilon^2)^(p/2)-f.u,
      // where u_x,u_y,u_z are the gradients of the scalar field u.
      // The state vector is defined as uu=[u_x,u_y,u_z,u].
 
      TDataType norm2 = uu[0] * uu[0] + uu[1] * uu[1] + uu[2] * uu[2];
      TDataType tvar = pow(ee * ee + norm2, (pp - 2.0) / 2.0);
 
      rr[0] = tvar * uu[0];
      rr[1] = tvar * uu[1];
      rr[2] = tvar * uu[2];
      rr[3] = -ff;
   }
};
 
/// Defines template class (functor) for evaluating the energy of the
/// p-Laplacian problem. The input parameters vparam are: vparam[0] - the
/// p-Laplacian power, vparam[1] small value ensuring exciting of an unique
/// solution, and vparam[2] - the distributed external input to the PDE. The
/// template parameter TDataType will be replaced by the compiler with the
/// appropriate AD type for automatic differentiation. The TParamVector
/// represents the vector type used for the parameter vector, and TStateVector
/// the vector type used for the state vector. The template parameters
/// state_size and param_size provide information for the size of the state and
/// the parameters vectors.
template<typename TDataType, typename TParamVector, typename TStateVector
         , int state_size, int param_size>
class MyEnergyFunctor
{
public:
   /// Returns the energy of a p-Laplacian for state field input provided in
   /// vector uu and parameters provided in vector vparam.
   TDataType operator()(TParamVector &vparam, TStateVector &uu)
   {
      MFEM_ASSERT(state_size==4,"MyEnergyFunctor state_size should be equal to 4!");
      MFEM_ASSERT(param_size==3,"MyEnergyFunctor param_size should be equal to 3!");
      real_t pp = vparam[0];
      real_t ee = vparam[1];
      real_t ff = vparam[2];
 
      TDataType u = uu[3];
      TDataType norm2 = uu[0] * uu[0] + uu[1] * uu[1] + uu[2] * uu[2];
 
      TDataType rez = pow(ee * ee + norm2, pp / 2.0) / pp - ff * u;
      return rez;
   }
};
 
 
/// Implements integrator for a p-Laplacian problem. The integrator is based on
/// a class QFunction utilized for evaluating the energy, the first derivative
/// (residual) and the Hessian of the energy (the Jacobian of the residual).
/// The template parameter CQVectAutoDiff represents the automatically
/// differentiated energy or residual implemented by the user.
/// CQVectAutoDiff::VectorFunc(Vector parameters, Vector state,Vector residual)
/// evaluates the residual at an integration point.
/// CQVectAutoDiff::Jacobian(Vector parameters, Vector state, Matrix hessian)
/// evaluates the Hessian of the energy(the Jacobian of the residual).
template<class CQVectAutoDiff>
class pLaplaceAD : public NonlinearFormIntegrator
{
protected:
   Coefficient *pp;
   Coefficient *coeff;
   Coefficient *load;
 
   CQVectAutoDiff rdf;
 
public:
   pLaplaceAD()
   {
      coeff = nullptr;
      pp = nullptr;
      load = nullptr;
 
      vparam.SetSize(3);
      vparam[0] = 2.0;  // default power
      vparam[1] = 1e-8; // default epsilon
      vparam[2] = 1.0;  // default load
   }
 
   pLaplaceAD(Coefficient &pp_) : pp(&pp_), coeff(nullptr), load(nullptr)
   {
      vparam.SetSize(3);
      vparam[0] = 2.0;  // default power
      vparam[1] = 1e-8; // default epsilon
      vparam[2] = 1.0;  // default load
 
   }
 
   pLaplaceAD(Coefficient &pp_, Coefficient &q, Coefficient &ld_)
      : pp(&pp_), coeff(&q), load(&ld_)
   {
      vparam.SetSize(3);
      vparam[0] = 2.0;  // default power
      vparam[1] = 1e-8; // default epsilon
      vparam[2] = 1.0;  // default load
   }
 
   virtual ~pLaplaceAD() {}
 
   virtual real_t GetElementEnergy(const FiniteElement &el,
                                   ElementTransformation &trans,
                                   const Vector &elfun)
   {
      real_t energy = 0.0;
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      // derivatives in isoparametric coordinates
      DenseMatrix dshape_iso(ndof, ndim);
      // derivatives in physical space
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
 
      Vector uu(4);     //[diff_x,diff_y,diff_z,u]
 
      uu = 0.0;
 
      // Calculates the functional/energy at an integration point.
      MyEnergyFunctor<real_t,Vector,Vector,4,3> qfunc;
 
      real_t w;
      real_t detJ;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
 
         // set the power
         if (pp != nullptr)
         {
            vparam[0] = pp->Eval(trans, ip);
         }
 
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            vparam[1] = coeff->Eval(trans, ip);
         }
         // add the contribution from the load
         if (load != nullptr)
         {
            vparam[2] = load->Eval(trans, ip);
         }
         // fill the values of vector uu
         for (int jj = 0; jj < spaceDim; jj++)
         {
            uu[jj] = grad[jj] / detJ;
         }
         uu[3] = shapef * elfun;
         // the energy is taken directly from the templated function
         energy = energy + w * qfunc(vparam,uu);
      }
      return energy;
   }
 
   virtual void AssembleElementVector(const FiniteElement &el,
                                      ElementTransformation &trans,
                                      const Vector &elfun,
                                      Vector &elvect)
   {
      MFEM_PERF_BEGIN("AssembleElementVector");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector lvec(ndof);
      elvect.SetSize(ndof);
      elvect = 0.0;
 
      DenseMatrix B(ndof, 4); // [diff_x,diff_y,diff_z, shape]
      Vector uu(4);           // [diff_x,diff_y,diff_z,u]
      Vector du(4);
      B = 0.0;
      uu = 0.0;
      real_t w;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         Mult(dshape_iso, trans.InverseJacobian(), dshape_xyz);
 
         // set the matrix B
         for (int jj = 0; jj < spaceDim; jj++)
         {
            B.SetCol(jj, dshape_xyz.GetColumn(jj));
         }
         B.SetCol(3, shapef);
 
         // set the power
         if (pp != nullptr)
         {
            vparam[0] = pp->Eval(trans, ip);
         }
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            vparam[1] = coeff->Eval(trans, ip);
         }
         // add the contribution from the load
         if (load != nullptr)
         {
            vparam[2] = load->Eval(trans, ip);
         }
 
         // calculate uu
         B.MultTranspose(elfun, uu);
         // calculate derivative of the energy with respect to uu
         rdf.VectorFunc(vparam,uu,du);
         B.Mult(du, lvec);
         elvect.Add(w, lvec);
      } // end integration loop
      MFEM_PERF_END("AssembleElementVector");
   }
 
   virtual void AssembleElementGrad(const FiniteElement &el,
                                    ElementTransformation &trans,
                                    const Vector &elfun,
                                    DenseMatrix &elmat)
   {
      MFEM_PERF_BEGIN("AssembleElementGrad");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      elmat.SetSize(ndof, ndof);
      elmat = 0.0;
 
      DenseMatrix B(ndof, 4); // [diff_x,diff_y,diff_z, shape]
      DenseMatrix A(ndof, 4);
      Vector uu(4); // [diff_x,diff_y,diff_z,u]
      DenseMatrix duu(4, 4);
      B = 0.0;
      uu = 0.0;
 
      real_t w;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         Mult(dshape_iso, trans.InverseJacobian(), dshape_xyz);
 
         // set the matrix B
         for (int jj = 0; jj < spaceDim; jj++)
         {
            B.SetCol(jj, dshape_xyz.GetColumn(jj));
         }
         B.SetCol(3, shapef);
 
         // set the power
         if (pp != nullptr)
         {
            vparam[0] = pp->Eval(trans, ip);
         }
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            vparam[1] = coeff->Eval(trans, ip);
         }
         // add the contribution from the load
         if (load != nullptr)
         {
            vparam[2] = load->Eval(trans, ip);
         }
 
         // calculate uu
         B.MultTranspose(elfun, uu);
         // calculate derivative of the energy with respect to uu
         rdf.Jacobian(vparam,uu,duu);
         Mult(B, duu, A);
         AddMult_a_ABt(w, A, B, elmat);
 
      } // end integration loop
      MFEM_PERF_END("AssembleElementGrad");
   }
 
private:
   Vector vparam; // [power, epsilon, load]
 
};
 
/// Implements hand-coded integrator for a p-Laplacian problem. Utilized as
/// alternative for the pLaplaceAD class based on automatic differentiation.
class pLaplace : public NonlinearFormIntegrator
{
protected:
   Coefficient *pp;
   Coefficient *coeff;
   Coefficient *load;
 
public:
   pLaplace()
   {
      coeff = nullptr;
      pp = nullptr;
      load = nullptr;
   }
 
   pLaplace(Coefficient &pp_) : pp(&pp_), coeff(nullptr), load(nullptr) {}
 
   pLaplace(Coefficient &pp_, Coefficient &q, Coefficient &ld_)
      : pp(&pp_), coeff(&q), load(&ld_)
   {}
 
   virtual ~pLaplace() {}
 
   virtual real_t GetElementEnergy(const FiniteElement &el,
                                   ElementTransformation &trans,
                                   const Vector &elfun)
   {
      real_t energy = 0.0;
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
 
      real_t w;
      real_t detJ;
      real_t nrgrad2;
      real_t ppp = 2.0;
      real_t eee = 0.0;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
         nrgrad2 = grad * grad / (detJ * detJ);
 
         // set the power
         if (pp != nullptr)
         {
            ppp = pp->Eval(trans, ip);
         }
 
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            eee = coeff->Eval(trans, ip);
         }
 
         energy = energy + w * std::pow(nrgrad2 + eee * eee, ppp / 2.0) / ppp;
 
         // add the contribution from the load
         if (load != nullptr)
         {
            energy = energy - w * (shapef * elfun) * load->Eval(trans, ip);
         }
      }
      return energy;
   }
 
   virtual void AssembleElementVector(const FiniteElement &el,
                                      ElementTransformation &trans,
                                      const Vector &elfun,
                                      Vector &elvect)
   {
      MFEM_PERF_BEGIN("AssembleElementVector");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
      Vector lvec(ndof);
      elvect.SetSize(ndof);
      elvect = 0.0;
 
      real_t w;
      real_t detJ;
      real_t nrgrad;
      real_t aa;
      real_t ppp = 2.0;
      real_t eee = 0.0;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
 
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
         nrgrad = grad.Norml2() / detJ;
         // grad is not scaled so far, i.e., grad=grad/detJ
 
         // set the power
         if (pp != nullptr)
         {
            ppp = pp->Eval(trans, ip);
         }
 
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            eee = coeff->Eval(trans, ip);
         }
         // compute (norm of the gradient)^2 + epsilon^2
         aa = nrgrad * nrgrad + eee * eee;
         aa = std::pow(aa, (ppp - 2.0) / 2.0);
         dshape_xyz.Mult(grad, lvec);
         elvect.Add(w * aa / (detJ * detJ), lvec);
 
         // add loading
         if (load != nullptr)
         {
            elvect.Add(-w * load->Eval(trans, ip), shapef);
         }
      } // end integration loop
      MFEM_PERF_END("AssembleElementVector");
   }
 
   virtual void AssembleElementGrad(const FiniteElement &el,
                                    ElementTransformation &trans,
                                    const Vector &elfun,
                                    DenseMatrix &elmat)
   {
      MFEM_PERF_BEGIN("AssembleElementGrad");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
      Vector lvec(ndof);
      // set the size of the element matrix
      elmat.SetSize(ndof, ndof);
      elmat = 0.0;
 
      real_t w; // integration weight
      real_t detJ;
      real_t nrgrad; // norm of the gradient
      real_t aa0; // original nonlinear diffusion coefficient
      real_t aa1; // gradient of the above
      real_t ppp = 2.0; // power in the P-Laplacian
      real_t eee = 0.0; // regularization parameter
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
         // grad is not scaled so far,i.e., grad=grad/detJ
 
         // set the power
         if (pp != nullptr)
         {
            ppp = pp->Eval(trans, ip);
         }
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            eee = coeff->Eval(trans, ip);
         }
 
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
         nrgrad = grad.Norml2() / detJ;
         // (u_x^2+u_y^2+u_z^2+\varepsilon^2)
         aa0 = nrgrad * nrgrad + eee * eee;
         aa1 = std::pow(aa0, (ppp - 2.0) / 2.0);
         aa0 = (ppp - 2.0) * std::pow(aa0, (ppp - 4.0) / 2.0);
         dshape_xyz.Mult(grad, lvec);
         w = w / (detJ * detJ);
         AddMult_a_VVt(w * aa0 / (detJ * detJ), lvec, elmat);
         AddMult_a_AAt(w * aa1, dshape_xyz, elmat);
 
      } // end integration loop
      MFEM_PERF_END("AssembleElementGrad");
   }
};
 
/// Implements AD enabled integrator for a p-Laplacian problem. The tangent
/// matrix is computed using the residual of the element. The template argument
/// should be equal to the size of the residual vector (element vector), i.e.,
/// the user should specify the size to match the exact vector size for the
/// considered order of the shape functions.
 
template<int sizeres=10>
class pLaplaceSL : public NonlinearFormIntegrator
{
protected:
   Coefficient *pp;
   Coefficient *coeff;
   Coefficient *load;
 
public:
   pLaplaceSL()
   {
      coeff = nullptr;
      pp = nullptr;
      load = nullptr;
   }
 
   pLaplaceSL(Coefficient &pp_) : pp(&pp_), coeff(nullptr), load(nullptr) {}
 
   pLaplaceSL(Coefficient &pp_, Coefficient &q, Coefficient &ld_)
      : pp(&pp_), coeff(&q), load(&ld_)
   {}
 
   virtual ~pLaplaceSL() {}
 
   virtual real_t GetElementEnergy(const FiniteElement &el,
                                   ElementTransformation &trans,
                                   const Vector &elfun)
   {
      real_t energy = 0.0;
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
 
      real_t w;
      real_t detJ;
      real_t nrgrad2;
      real_t ppp = 2.0;
      real_t eee = 0.0;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
         nrgrad2 = grad * grad / (detJ * detJ);
 
         // set the power
         if (pp != nullptr)
         {
            ppp = pp->Eval(trans, ip);
         }
 
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            eee = coeff->Eval(trans, ip);
         }
 
         energy = energy + w * std::pow(nrgrad2 + eee * eee, ppp / 2.0) / ppp;
 
         // add the contribution from the load
         if (load != nullptr)
         {
            energy = energy - w * (shapef * elfun) * load->Eval(trans, ip);
         }
      }
      return energy;
   }
 
   virtual void AssembleElementVector(const FiniteElement &el,
                                      ElementTransformation &trans,
                                      const Vector &elfun,
                                      Vector &elvect)
   {
      MFEM_PERF_BEGIN("AssembleElementVector");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      Vector shapef(ndof);
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      Vector grad(spaceDim);
      Vector lvec(ndof);
      elvect.SetSize(ndof);
      elvect = 0.0;
 
      real_t w;
      real_t detJ;
      real_t nrgrad;
      real_t aa;
      real_t ppp = 2.0;
      real_t eee = 0.0;
 
      for (int i = 0; i < ir.GetNPoints(); i++)
      {
         const IntegrationPoint &ip = ir.IntPoint(i);
         trans.SetIntPoint(&ip);
         w = trans.Weight();
         detJ = (square ? w : w * w);
         w = ip.weight * w; //w;
 
         el.CalcDShape(ip, dshape_iso);
         el.CalcShape(ip, shapef);
         // AdjugateJacobian = / adj(J),         if J is square
         //                    \ adj(J^t.J).J^t, otherwise
         Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
         // dshape_xyz should be divided by detJ for obtaining the real value
 
         // calculate the gradient
         dshape_xyz.MultTranspose(elfun, grad);
         nrgrad = grad.Norml2() / detJ;
         // grad is not scaled so far, i.e., grad=grad/detJ
 
         // set the power
         if (pp != nullptr)
         {
            ppp = pp->Eval(trans, ip);
         }
 
         // set the coefficient ensuring positiveness of the tangent matrix
         if (coeff != nullptr)
         {
            eee = coeff->Eval(trans, ip);
         }
 
         aa = nrgrad * nrgrad + eee * eee;
         aa = std::pow(aa, (ppp - 2.0) / 2.0);
         dshape_xyz.Mult(grad, lvec);
         elvect.Add(w * aa / (detJ * detJ), lvec);
 
         // add loading
         if (load != nullptr)
         {
            elvect.Add(-w * load->Eval(trans, ip), shapef);
         }
      } // end integration loop
      MFEM_PERF_END("AssembleElementVector");
   }
 
   virtual void AssembleElementGrad(const FiniteElement &el,
                                    ElementTransformation &trans,
                                    const Vector &elfun,
                                    DenseMatrix &elmat)
   {
      MFEM_PERF_BEGIN("AssembleElementGrad");
      const int ndof = el.GetDof();
      const int ndim = el.GetDim();
      const int spaceDim = trans.GetSpaceDim();
      bool square = (ndim == spaceDim);
      int order = 2 * el.GetOrder() + trans.OrderGrad(&el);
      const IntegrationRule &ir(IntRules.Get(el.GetGeomType(), order));
 
      DenseMatrix dshape_iso(ndof, ndim);
      DenseMatrix dshape_xyz(ndof, spaceDim);
      elmat.SetSize(ndof, ndof);
      elmat = 0.0;
 
      real_t w;
      real_t detJ;
      real_t ppp = 2.0;
      real_t eee = 0.0;
 
      mfem::Vector param(3); param=0.0;
 
      // Computes the residual at an integration point. The implementation is a
      // copy of the integration loop in AssembleElementVector.
      auto resfun = [&](mfem::Vector& vparam, mfem::ad::ADVectorType& uu,
                        mfem::ad::ADVectorType& vres)
      {
 
         vres.SetSize(uu.Size()); vres=0.0;
         mfem::ad::ADVectorType grad(spaceDim);
         mfem::ad::ADFloatType nrgrad;
         mfem::ad::ADFloatType aa;
         mfem::ad::ADVectorType lvec(ndof);
 
         for (int q = 0; q < ir.GetNPoints(); q++)
         {
            lvec=0.0;
 
            const IntegrationPoint &ip = ir.IntPoint(q);
            trans.SetIntPoint(&ip);
            w = trans.Weight();
            detJ = (square ? w : w * w);
            w = ip.weight * w;
 
            el.CalcDShape(ip, dshape_iso);
            // AdjugateJacobian = / adj(J),         if J is square
            //                    \ adj(J^t.J).J^t, otherwise
            Mult(dshape_iso, trans.AdjugateJacobian(), dshape_xyz);
            // dshape_xyz should be divided by detJ for obtaining the real value
            // grad is not scaled so far,i.e., grad=grad/detJ
 
            // set the power
            if (pp != nullptr)
            {
               ppp = pp->Eval(trans, ip);
            }
            // set the coefficient ensuring positiveness of the tangent matrix
            if (coeff != nullptr)
            {
               eee = coeff->Eval(trans, ip);
            }
 
            grad=0.0;
            // calculate the gradient
            for (int i=0; i<spaceDim; i++)
            {
               for (int j=0; j<ndof; j++)
               {
                  grad[i]= grad[i]+ dshape_xyz(j,i)*uu[j];
               }
            }
 
 
            nrgrad= (grad*grad)/(detJ*detJ);
 
            aa = nrgrad + eee * eee;
            aa = pow(aa, (ppp - 2.0) / 2.0);
 
            for (int i=0; i<spaceDim; i++)
            {
               for (int j=0; j<ndof; j++)
               {
                  lvec[j] = lvec[j] + dshape_xyz(j,i) * grad[i];
               }
            }
 
            for (int j=0; j<ndof; j++)
            {
               vres[j]=vres[j] + lvec[j] * (w*aa/(detJ*detJ));
            }
         }
      };
 
      mfem::Vector bla(elfun);
      // calculate the gradient - only for a fixed ndof
      mfem::VectorFuncAutoDiff<sizeres,sizeres,3> fdr(resfun);
      fdr.Jacobian(param, bla, elmat);
      MFEM_PERF_END("AssembleElementGrad");
   }
};
 
} // namespace mfem
 
#endif