GenericProblem.cpp

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#include "GenericProblem.hpp"
 
#include <polyfem/utils/JSONUtils.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/StringUtils.hpp>
#include <polyfem/io/MatrixIO.hpp>
 
namespace polyfem
{
    using namespace utils;
 
    namespace assembler
    {
        namespace
        {
            std::vector<json> flatten_ids(const json &p_j_boundary_tmp)
            {
                const std::vector<json> j_boundary_tmp = utils::json_as_array(p_j_boundary_tmp);
 
                std::vector<json> j_boundary;
 
                for (size_t i = 0; i < j_boundary_tmp.size(); ++i)
                {
                    const auto &tmp = j_boundary_tmp[i];
 
                    if (tmp.is_string())
                    {
                        j_boundary.push_back(tmp);
                        continue;
                    }
                    if (!tmp.contains("id"))
                        continue;
 
                    if (tmp["id"].is_array())
                    {
                        for (size_t j = 0; j < tmp["id"].size(); ++j)
                        {
                            json newj = tmp;
                            newj["id"] = tmp["id"][j].get<int>();
                            j_boundary.push_back(newj);
                        }
                    }
                    else
                        j_boundary.push_back(tmp);
                }
 
                return j_boundary;
            }
        } // namespace
 
        double TensorBCValue::eval(const RowVectorNd &pts, const int dim, const double t, const int el_id) const
        {
            double x = pts(0);
            double y = pts(1);
            double z = pts.size() == 2 ? 0 : pts(2);
 
            double val = value[dim](x, y, z, t, el_id);
 
            if (interpolation.empty())
            {
            }
            else if (interpolation.size() == 1)
                val *= interpolation[0]->eval(t);
            else
            {
                assert(dim < interpolation.size());
                val *= interpolation[dim]->eval(t);
            }
 
            return val;
        }
 
        double ScalarBCValue::eval(const RowVectorNd &pts, const double t) const
        {
            assert(pts.size() == 2 || pts.size() == 3);
            double x = pts(0), y = pts(1), z = pts.size() == 3 ? pts(2) : 0.0;
            return value(x, y, z, t) * interpolation->eval(t);
        }
 
        GenericTensorProblem::GenericTensorProblem(const std::string &name)
            : Problem(name), is_all_(false)
        {
        }
 
        void GenericTensorProblem::set_units(const assembler::Assembler &assembler, const Units &units)
        {
            if (assembler.is_fluid())
            {
                // TODO
                assert(false);
            }
            else
            {
                for (int i = 0; i < 3; ++i)
                {
                    rhs_[i].set_unit_type(units.acceleration());
                    exact_[i].set_unit_type(units.length());
                }
                for (int i = 0; i < 3; ++i)
                    exact_grad_[i].set_unit_type("");
 
                for (auto &v : displacements_)
                    v.set_unit_type(units.length());
 
                for (auto &v : forces_)
                    v.set_unit_type(units.force());
 
                for (auto &v : normal_aligned_forces_)
                    v.set_unit_type(units.pressure());
 
                for (auto &v : pressures_)
                    v.set_unit_type(units.pressure());
 
                for (auto &v : cavity_pressures_)
                    v.second.set_unit_type(units.pressure());
 
                for (auto &v : initial_position_)
                    for (int i = 0; i < 3; ++i)
                        v.second[i].set_unit_type(units.length());
 
                for (auto &v : initial_velocity_)
                    for (int i = 0; i < 3; ++i)
                        v.second[i].set_unit_type(units.velocity());
 
                for (auto &v : initial_acceleration_)
                    for (int i = 0; i < 3; ++i)
                        v.second[i].set_unit_type(units.acceleration());
 
                for (auto &v : nodal_dirichlet_)
                    v.second.set_unit_type(units.length());
 
                for (auto &v : nodal_neumann_)
                    v.second.set_unit_type(units.force());
            }
        }
 
        void GenericTensorProblem::rhs(const assembler::Assembler &assembler, const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), pts.cols());
 
            if (is_rhs_zero())
            {
                val.setZero();
                return;
            }
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                for (int j = 0; j < pts.cols(); ++j)
                {
                    double x = pts(i, 0), y = pts(i, 1), z = planar ? 0 : pts(i, 2);
                    val(i, j) = rhs_[j](x, y, z, t);
                }
            }
        }
 
        bool GenericTensorProblem::is_dimension_dirichet(const int tag, const int dim) const
        {
            if (all_dimensions_dirichlet())
                return true;
 
            if (is_all_)
            {
                assert(displacements_.size() == 1);
                return displacements_[0].dirichlet_dimension[dim];
            }
 
            for (size_t b = 0; b < boundary_ids_.size(); ++b)
            {
                if (tag == boundary_ids_[b])
                {
                    auto &tmp = displacements_[b].dirichlet_dimension;
                    return tmp[dim];
                }
            }
 
            assert(false);
            return true;
        }
 
        void GenericTensorProblem::dirichlet_bc(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &uv, const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(pts.rows(), mesh.dimension());
 
            for (long i = 0; i < pts.rows(); ++i)
            {
                if (is_all_)
                {
                    assert(displacements_.size() == 1);
                    for (int d = 0; d < val.cols(); ++d)
                    {
                        val(i, d) = displacements_[0].eval(pts.row(i), d, t);
                    }
                }
                else
                {
                    const int id = mesh.get_boundary_id(global_ids(i));
                    for (size_t b = 0; b < boundary_ids_.size(); ++b)
                    {
                        if (id == boundary_ids_[b])
                        {
                            for (int d = 0; d < val.cols(); ++d)
                            {
                                val(i, d) = displacements_[b].eval(pts.row(i), d, t);
                            }
 
                            break;
                        }
                    }
                }
            }
        }
 
        void GenericTensorProblem::neumann_bc(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &uv, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &normals, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(pts.rows(), mesh.dimension());
 
            for (long i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_boundary_id(global_ids(i));
 
                for (size_t b = 0; b < neumann_boundary_ids_.size(); ++b)
                {
                    if (id == neumann_boundary_ids_[b])
                    {
                        for (int d = 0; d < val.cols(); ++d)
                        {
                            val(i, d) = forces_[b].eval(pts.row(i), d, t);
                        }
 
                        break;
                    }
                }
 
                for (size_t b = 0; b < normal_aligned_neumann_boundary_ids_.size(); ++b)
                {
                    if (id == normal_aligned_neumann_boundary_ids_[b])
                    {
                        for (int d = 0; d < val.cols(); ++d)
                        {
                            val(i, d) = normal_aligned_forces_[b].eval(pts.row(i), t) * normals(i, d);
                        }
                        break;
                    }
                }
            }
        }
 
        void GenericTensorProblem::pressure_bc(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &uv, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &normals, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(pts.rows(), 1);
 
            for (long i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_boundary_id(global_ids(i));
 
                for (size_t b = 0; b < pressure_boundary_ids_.size(); ++b)
                {
                    if (id == pressure_boundary_ids_[b])
                    {
                        val(i) = pressures_[b].eval(pts.row(i), t);
                        break;
                    }
                }
            }
        }
 
        double GenericTensorProblem::pressure_cavity_bc(const int boundary_id, const double t) const
        {
            Eigen::VectorXd pt;
            pt.setZero(3);
            return cavity_pressures_.at(boundary_id).eval(pt, t);
        }
 
        void GenericTensorProblem::exact(const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            assert(has_exact_sol());
            const bool planar = pts.cols() == 2;
            val.resize(pts.rows(), pts.cols());
 
            for (int i = 0; i < pts.rows(); ++i)
            {
                for (int j = 0; j < pts.cols(); ++j)
                {
                    double x = pts(i, 0), y = pts(i, 1), z = planar ? 0 : pts(i, 2);
                    val(i, j) = exact_[j](x, y, z, t);
                }
            }
        }
 
        void GenericTensorProblem::exact_grad(const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            const int size = pts.cols();
            val.resize(pts.rows(), pts.cols() * size);
            if (!has_exact_grad_)
                return;
 
            const bool planar = size == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                for (int j = 0; j < pts.cols() * size; ++j)
                {
                    double x = pts(i, 0), y = pts(i, 1), z = planar ? 0 : pts(i, 2);
                    val(i, j) = exact_grad_[j](x, y, z, t);
                }
            }
        }
 
        void GenericTensorProblem::add_dirichlet_boundary(const int id, const Eigen::RowVector3d &val, const bool isx, const bool isy, const bool isz, const std::shared_ptr<Interpolation> &interp)
        {
            boundary_ids_.push_back(id);
 
            displacements_.emplace_back();
            for (size_t k = 0; k < val.size(); ++k)
                displacements_.back().value[k].init(val[k]);
 
            displacements_.back().dirichlet_dimension << isx, isy, isz;
            displacements_.back().interpolation.push_back(interp);
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::update_dirichlet_boundary(const int id, const Eigen::RowVector3d &val, const bool isx, const bool isy, const bool isz, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            displacements_[index].interpolation.clear();
            displacements_[index].interpolation.push_back(interp);
 
            for (size_t k = 0; k < val.size(); ++k)
                displacements_[index].value[k].init(val[k]);
 
            displacements_[index].dirichlet_dimension << isx, isy, isz;
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::add_neumann_boundary(const int id, const Eigen::RowVector3d &val, const std::shared_ptr<Interpolation> &interp)
        {
            neumann_boundary_ids_.push_back(id);
            forces_.emplace_back();
            for (size_t k = 0; k < val.size(); ++k)
                forces_.back().value[k].init(val[k]);
 
            forces_.back().interpolation.push_back(interp);
        }
 
        void GenericTensorProblem::update_neumann_boundary(const int id, const Eigen::RowVector3d &val, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            forces_[index].interpolation.clear();
            forces_[index].interpolation.push_back(interp);
            for (size_t k = 0; k < val.size(); ++k)
                forces_[index].value[k].init(val[k]);
        }
 
        void GenericTensorProblem::add_pressure_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            pressure_boundary_ids_.push_back(id);
            pressures_.emplace_back();
            pressures_.back().value.init(val);
            pressures_.back().interpolation = interp;
        }
 
        void GenericTensorProblem::update_pressure_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < pressure_boundary_ids_.size(); ++i)
            {
                if (pressure_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            pressures_[index].interpolation = interp;
            pressures_[index].value.init(val);
        }
 
        void GenericTensorProblem::add_dirichlet_boundary(const int id, const std::function<Eigen::MatrixXd(double x, double y, double z, double t)> &func, const bool isx, const bool isy, const bool isz, const std::shared_ptr<Interpolation> &interp)
        {
            boundary_ids_.push_back(id);
            displacements_.emplace_back();
            displacements_.back().interpolation.push_back(interp);
            for (size_t k = 0; k < displacements_.back().value.size(); ++k)
                displacements_.back().value[k].init(func, k);
 
            displacements_.back().dirichlet_dimension << isx, isy, isz;
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::update_dirichlet_boundary(const int id, const std::function<Eigen::MatrixXd(double x, double y, double z, double t)> &func, const bool isx, const bool isy, const bool isz, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            displacements_[index].interpolation.clear();
            displacements_[index].interpolation.push_back(interp);
            for (size_t k = 0; k < displacements_.back().value.size(); ++k)
                displacements_[index].value[k].init(func, k);
 
            displacements_[index].dirichlet_dimension << isx, isy, isz;
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::add_neumann_boundary(const int id, const std::function<Eigen::MatrixXd(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            neumann_boundary_ids_.push_back(id);
            forces_.emplace_back();
            forces_.back().interpolation.push_back(interp);
            for (size_t k = 0; k < forces_.back().value.size(); ++k)
                forces_.back().value[k].init(func, k);
        }
 
        void GenericTensorProblem::update_neumann_boundary(const int id, const std::function<Eigen::MatrixXd(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            forces_[index].interpolation.clear();
            forces_[index].interpolation.push_back(interp);
            for (size_t k = 0; k < forces_.back().value.size(); ++k)
                forces_[index].value[k].init(func, k);
        }
 
        void GenericTensorProblem::add_pressure_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            pressure_boundary_ids_.push_back(id);
            pressures_.emplace_back();
            pressures_.back().value.init(func);
            pressures_.back().interpolation = interp;
        }
 
        void GenericTensorProblem::update_pressure_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < pressure_boundary_ids_.size(); ++i)
            {
                if (pressure_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            pressures_[index].interpolation = interp;
            pressures_[index].value.init(func);
        }
 
        void GenericTensorProblem::add_dirichlet_boundary(const int id, const json &val, const bool isx, const bool isy, const bool isz, const std::string &interpolation)
        {
            if (!val.is_array())
                throw "Val must be an array";
 
            boundary_ids_.push_back(id);
            displacements_.emplace_back();
            if (!interpolation.empty())
                displacements_.back().interpolation.push_back(Interpolation::build(interpolation));
 
            for (size_t k = 0; k < val.size(); ++k)
                displacements_.back().value[k].init(val[k]);
 
            displacements_.back().dirichlet_dimension << isx, isy, isz;
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::add_neumann_boundary(const int id, const json &val, const std::string &interpolation)
        {
            if (!val.is_array())
                throw "Val must be an array";
 
            neumann_boundary_ids_.push_back(id);
 
            forces_.emplace_back();
            if (!interpolation.empty())
                forces_.back().interpolation.push_back(Interpolation::build(interpolation));
 
            for (size_t k = 0; k < val.size(); ++k)
                forces_.back().value[k].init(val[k]);
        }
 
        void GenericTensorProblem::add_pressure_boundary(const int id, json val, const std::string &interpolation)
        {
            pressure_boundary_ids_.push_back(id);
            pressures_.emplace_back();
 
            if (interpolation.empty())
                pressures_.back().interpolation = std::make_shared<NoInterpolation>();
            else
                pressures_.back().interpolation = Interpolation::build(interpolation);
            pressures_.back().value.init(val);
        }
 
        void GenericTensorProblem::update_dirichlet_boundary(const int id, const json &val, const bool isx, const bool isy, const bool isz, const std::string &interpolation)
        {
            if (!val.is_array())
                throw "Val must be an array";
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            displacements_[index].interpolation.clear();
            if (!interpolation.empty())
                displacements_[index].interpolation.push_back(Interpolation::build(interpolation));
 
            for (size_t k = 0; k < val.size(); ++k)
                displacements_[index].value[k].init(val[k]);
 
            displacements_[index].dirichlet_dimension << isx, isy, isz;
 
            if (!isx || !isy || !isz)
                all_dimensions_dirichlet_ = false;
        }
 
        void GenericTensorProblem::update_neumann_boundary(const int id, const json &val, const std::string &interpolation)
        {
            if (!val.is_array())
                throw "Val must be an array";
 
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            forces_[index].interpolation.clear();
            if (!interpolation.empty())
                forces_[index].interpolation.push_back(Interpolation::build(interpolation));
 
            for (size_t k = 0; k < val.size(); ++k)
                forces_[index].value[k].init(val[k]);
        }
 
        void GenericTensorProblem::update_pressure_boundary(const int id, json val, const std::string &interpolation)
        {
            int index = -1;
            for (int i = 0; i < pressure_boundary_ids_.size(); ++i)
            {
                if (pressure_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            if (interpolation.empty())
                pressures_[index].interpolation = std::make_shared<NoInterpolation>();
            else
                pressures_[index].interpolation = Interpolation::build(interpolation);
            pressures_[index].value.init(val);
        }
 
        void GenericTensorProblem::set_rhs(double x, double y, double z)
        {
            rhs_[0].init(x);
            rhs_[1].init(y);
            rhs_[2].init(z);
        }
 
        void GenericTensorProblem::dirichlet_nodal_value(const mesh::Mesh &mesh, const int node_id, const RowVectorNd &pt, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(1, mesh.dimension());
            const int tag = mesh.get_node_id(node_id);
 
            if (is_all_)
            {
                assert(nodal_dirichlet_.size() == 1);
                const auto &tmp = nodal_dirichlet_.begin()->second;
 
                for (int d = 0; d < val.cols(); ++d)
                {
                    val(d) = tmp.eval(pt, d, t);
                }
 
                return;
            }
 
            const auto it = nodal_dirichlet_.find(tag);
            if (it != nodal_dirichlet_.end())
            {
                for (int d = 0; d < val.cols(); ++d)
                {
                    val(d) = it->second.eval(pt, d, t);
                }
 
                return;
            }
 
            for (const auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int i = 0; i < n_dirichlet.rows(); ++i)
                {
                    if (n_dirichlet(i, 0) == node_id)
                    {
                        for (int d = 0; d < val.cols(); ++d)
                        {
                            val(d) = n_dirichlet(i, d + 1);
                        }
 
                        return;
                    }
                }
            }
 
            assert(false);
        }
 
        void GenericTensorProblem::neumann_nodal_value(const mesh::Mesh &mesh, const int node_id, const RowVectorNd &pt, const Eigen::MatrixXd &normal, const double t, Eigen::MatrixXd &val) const
        {
            // TODO implement me;
            log_and_throw_error("Nodal neumann not implemented");
        }
 
        bool GenericTensorProblem::is_nodal_dirichlet_boundary(const int n_id, const int tag)
        {
            if (nodal_dirichlet_.find(tag) != nodal_dirichlet_.end())
                return true;
 
            for (const auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int i = 0; i < n_dirichlet.rows(); ++i)
                {
                    if (n_dirichlet(i, 0) == n_id)
                        return true;
                }
            }
 
            return false;
        }
 
        bool GenericTensorProblem::is_nodal_neumann_boundary(const int n_id, const int tag)
        {
            return nodal_neumann_.find(tag) != nodal_neumann_.end();
        }
 
        bool GenericTensorProblem::has_nodal_dirichlet()
        {
            return nodal_dirichlet_mat_.size() > 0;
        }
 
        bool GenericTensorProblem::has_nodal_neumann()
        {
            return false; // nodal_neumann_.size() > 0;
        }
 
        bool GenericTensorProblem::is_nodal_dimension_dirichlet(const int n_id, const int tag, const int dim) const
        {
            const auto it = nodal_dirichlet_.find(tag);
            if (it != nodal_dirichlet_.end())
            {
                return it->second.dirichlet_dimension(dim);
            }
 
            for (const auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int i = 0; i < n_dirichlet.rows(); ++i)
                {
                    if (n_dirichlet(i, 0) == n_id)
                    {
                        return !std::isnan(n_dirichlet(i, dim + 1));
                    }
                }
            }
 
            assert(false);
            return true;
        }
 
        void GenericTensorProblem::update_nodes(const Eigen::VectorXi &in_node_to_node)
        {
            for (auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int n = 0; n < n_dirichlet.rows(); ++n)
                {
                    const int node_id = in_node_to_node[n_dirichlet(n, 0)];
                    n_dirichlet(n, 0) = node_id;
                }
            }
        }
 
        void GenericTensorProblem::set_parameters(const json &params)
        {
            if (is_param_valid(params, "is_time_dependent"))
            {
                is_time_dept_ = params["is_time_dependent"];
            }
 
            if (is_param_valid(params, "rhs"))
            {
                auto rr = params["rhs"];
                if (rr.is_array())
                {
                    for (size_t k = 0; k < rr.size(); ++k)
                        rhs_[k].init(rr[k]);
                }
                else
                {
                    logger().warn("Invalid problem rhs: should be an array.");
                    assert(false);
                }
            }
 
            if (is_param_valid(params, "reference") && is_param_valid(params["reference"], "solution"))
            {
                auto ex = params["reference"]["solution"];
                has_exact_ = ex.size() > 0;
                if (ex.is_array())
                {
                    for (size_t k = 0; k < ex.size(); ++k)
                        exact_[k].init(ex[k]);
                }
                else
                {
                    assert(false);
                }
            }
 
            if (is_param_valid(params, "reference") && is_param_valid(params["reference"], "gradient"))
            {
                auto ex = params["reference"]["gradient"];
                has_exact_grad_ = ex.size() > 0;
                if (ex.is_array())
                {
                    for (size_t k = 0; k < ex.size(); ++k)
                        exact_grad_[k].init(ex[k]);
                }
                else
                {
                    assert(false);
                }
            }
 
            if (is_param_valid(params, "dirichlet_boundary"))
            {
                // boundary_ids_.clear();
                int offset = boundary_ids_.size();
                std::vector<json> j_boundary = flatten_ids(params["dirichlet_boundary"]);
 
                boundary_ids_.resize(offset + j_boundary.size());
                displacements_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < boundary_ids_.size(); ++i)
                {
                    if (j_boundary[i - offset].is_string())
                    {
                        const std::string path = resolve_path(j_boundary[i - offset], params["root_path"]);
                        if (!std::filesystem::is_regular_file(path))
                            log_and_throw_error("unable to open {} file", path);
 
                        Eigen::MatrixXd tmp;
                        io::read_matrix(path, tmp);
                        nodal_dirichlet_mat_.emplace_back(tmp);
 
                        continue;
                    }
 
                    int current_id = -1;
 
                    if (j_boundary[i - offset]["id"] == "all")
                    {
                        assert(boundary_ids_.size() == 1);
                        boundary_ids_.clear();
                        is_all_ = true;
                        nodal_dirichlet_[current_id] = TensorBCValue();
                    }
                    else
                    {
                        boundary_ids_[i] = j_boundary[i - offset]["id"];
                        current_id = boundary_ids_[i];
                        nodal_dirichlet_[current_id] = TensorBCValue();
                    }
 
                    auto ff = j_boundary[i - offset]["value"];
                    if (ff.is_array())
                    {
                        for (size_t k = 0; k < ff.size(); ++k)
                        {
                            displacements_[i].value[k].init(ff[k]);
                            if (j_boundary[i - offset].contains("time_reference") && j_boundary[i - offset]["time_reference"].size() > 0)
                                displacements_[i].value[k].set_t(j_boundary[i - offset]["time_reference"]);
                            nodal_dirichlet_[current_id].value[k].init(ff[k]);
                        }
                    }
                    else
                    {
                        assert(false);
                        displacements_[i].value[0].init(0);
                        displacements_[i].value[1].init(0);
                        displacements_[i].value[2].init(0);
                    }
 
                    displacements_[i].dirichlet_dimension.setConstant(true);
                    nodal_dirichlet_[current_id].dirichlet_dimension.setConstant(true);
                    if (j_boundary[i - offset].contains("dimension"))
                    {
                        all_dimensions_dirichlet_ = false;
                        auto &tmp = j_boundary[i - offset]["dimension"];
                        assert(tmp.is_array());
                        for (size_t k = 0; k < tmp.size(); ++k)
                        {
                            displacements_[i].dirichlet_dimension[k] = tmp[k];
                            nodal_dirichlet_[current_id].dirichlet_dimension[k] = tmp[k];
                        }
                    }
 
                    if (j_boundary[i - offset]["interpolation"].is_array())
                    {
                        for (int ii = 0; ii < j_boundary[i - offset]["interpolation"].size(); ++ii)
                            displacements_[i].interpolation.push_back(Interpolation::build(j_boundary[i - offset]["interpolation"][ii]));
                    }
                    else
                        displacements_[i].interpolation.push_back(Interpolation::build(j_boundary[i - offset]["interpolation"]));
 
                    nodal_dirichlet_[current_id].interpolation = displacements_[i].interpolation;
                }
            }
 
            if (is_param_valid(params, "neumann_boundary"))
            {
                // neumann_boundary_ids_.clear();
                const int offset = neumann_boundary_ids_.size();
 
                auto j_boundary_tmp = params["neumann_boundary"];
                std::vector<json> j_boundary = flatten_ids(j_boundary_tmp);
 
                neumann_boundary_ids_.resize(offset + j_boundary.size());
                forces_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < neumann_boundary_ids_.size(); ++i)
                {
                    neumann_boundary_ids_[i] = j_boundary[i - offset]["id"];
 
                    auto ff = j_boundary[i - offset]["value"];
                    assert(ff.is_array());
 
                    for (size_t k = 0; k < ff.size(); ++k)
                        forces_[i].value[k].init(ff[k]);
 
                    if (j_boundary[i - offset]["interpolation"].is_array())
                    {
                        for (int ii = 0; ii < j_boundary[i - offset]["interpolation"].size(); ++ii)
                            forces_[i].interpolation.push_back(Interpolation::build(j_boundary[i - offset]["interpolation"][ii]));
                    }
                    else
                    {
                        forces_[i].interpolation.push_back(Interpolation::build(j_boundary[i - offset]["interpolation"]));
                    }
                }
            }
 
            if (is_param_valid(params, "normal_aligned_neumann_boundary"))
            {
                const int offset = normal_aligned_neumann_boundary_ids_.size();
 
                auto j_boundary_tmp = params["normal_aligned_neumann_boundary"];
                std::vector<json> j_boundary = flatten_ids(j_boundary_tmp);
 
                normal_aligned_neumann_boundary_ids_.resize(offset + j_boundary.size());
                normal_aligned_forces_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < normal_aligned_neumann_boundary_ids_.size(); ++i)
                {
                    normal_aligned_neumann_boundary_ids_[i] = j_boundary[i - offset]["id"];
 
                    auto ff = j_boundary[i - offset]["value"];
                    normal_aligned_forces_[i].value.init(ff);
 
                    if (j_boundary[i - offset].contains("interpolation"))
                        normal_aligned_forces_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"]);
                    else
                        normal_aligned_forces_[i].interpolation = std::make_shared<NoInterpolation>();
                }
            }
 
            if (is_param_valid(params, "pressure_boundary"))
            {
                // pressure_boundary_ids_.clear();
                const int offset = pressure_boundary_ids_.size();
 
                auto j_boundary_tmp = params["pressure_boundary"];
                std::vector<json> j_boundary = flatten_ids(j_boundary_tmp);
 
                pressure_boundary_ids_.resize(offset + j_boundary.size());
                pressures_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < pressure_boundary_ids_.size(); ++i)
                {
                    pressure_boundary_ids_[i] = j_boundary[i - offset]["id"];
 
                    auto ff = j_boundary[i - offset]["value"];
                    pressures_[i].value.init(ff);
 
                    if (j_boundary[i - offset].contains("interpolation"))
                        pressures_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"]);
                    else
                        pressures_[i].interpolation = std::make_shared<NoInterpolation>();
                }
            }
 
            if (is_param_valid(params, "pressure_cavity"))
            {
                const int offset = pressure_cavity_ids_.size();
 
                auto j_boundary_tmp = params["pressure_cavity"];
                std::vector<json> j_boundary = flatten_ids(j_boundary_tmp);
 
                pressure_cavity_ids_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < pressure_cavity_ids_.size(); ++i)
                {
                    int boundary_id = j_boundary[i - offset]["id"];
                    pressure_cavity_ids_[i] = boundary_id;
 
                    if (cavity_pressures_.find(boundary_id) == cavity_pressures_.end())
                    {
                        cavity_pressures_[boundary_id] = ScalarBCValue();
 
                        auto ff = j_boundary[i - offset]["value"];
                        cavity_pressures_[boundary_id].value.init(ff);
                    }
                }
            }
 
            if (is_param_valid(params, "solution"))
            {
                auto rr = params["solution"];
                initial_position_.resize(rr.size());
                assert(rr.is_array());
 
                for (size_t k = 0; k < rr.size(); ++k)
                {
                    initial_position_[k].first = rr[k]["id"];
                    const auto v = rr[k]["value"];
                    for (size_t d = 0; d < v.size(); ++d)
                        initial_position_[k].second[d].init(v[d]);
                }
            }
 
            if (is_param_valid(params, "velocity"))
            {
                auto rr = params["velocity"];
                initial_velocity_.resize(rr.size());
                assert(rr.is_array());
 
                for (size_t k = 0; k < rr.size(); ++k)
                {
                    initial_velocity_[k].first = rr[k]["id"];
                    const auto v = rr[k]["value"];
                    for (size_t d = 0; d < v.size(); ++d)
                        initial_velocity_[k].second[d].init(v[d]);
                }
            }
 
            if (is_param_valid(params, "acceleration"))
            {
                auto rr = params["acceleration"];
                initial_acceleration_.resize(rr.size());
                assert(rr.is_array());
 
                for (size_t k = 0; k < rr.size(); ++k)
                {
                    initial_acceleration_[k].first = rr[k]["id"];
                    const auto v = rr[k]["value"];
                    for (size_t d = 0; d < v.size(); ++d)
                        initial_acceleration_[k].second[d].init(v[d]);
                }
            }
        }
 
        void GenericTensorProblem::initial_solution(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &pts, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), pts.cols());
            if (initial_position_.empty())
            {
                val.setZero();
                return;
            }
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_body_id(global_ids(i));
                int index = -1;
                for (int j = 0; j < initial_position_.size(); ++j)
                {
                    if (initial_position_[j].first == id)
                    {
                        index = j;
                        break;
                    }
                }
                if (index < 0)
                {
                    val.row(i).setZero();
                    continue;
                }
 
                for (int j = 0; j < pts.cols(); ++j)
                    val(i, j) = planar ? initial_position_[index].second[j](pts(i, 0), pts(i, 1)) : initial_position_[index].second[j](pts(i, 0), pts(i, 1), pts(i, 2));
            }
        }
 
        void GenericTensorProblem::initial_velocity(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &pts, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), pts.cols());
            if (initial_velocity_.empty())
            {
                val.setZero();
                return;
            }
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_body_id(global_ids(i));
                int index = -1;
                for (int j = 0; j < initial_velocity_.size(); ++j)
                {
                    if (initial_velocity_[j].first == id)
                    {
                        index = j;
                        break;
                    }
                }
                if (index < 0)
                {
                    val.row(i).setZero();
                    continue;
                }
 
                for (int j = 0; j < pts.cols(); ++j)
                    val(i, j) = planar ? initial_velocity_[index].second[j](pts(i, 0), pts(i, 1)) : initial_velocity_[index].second[j](pts(i, 0), pts(i, 1), pts(i, 2));
            }
        }
 
        void GenericTensorProblem::initial_acceleration(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &pts, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), pts.cols());
            if (initial_acceleration_.empty())
            {
                val.setZero();
                return;
            }
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_body_id(global_ids(i));
                int index = -1;
                for (int j = 0; j < initial_acceleration_.size(); ++j)
                {
                    if (initial_acceleration_[j].first == id)
                    {
                        index = j;
                        break;
                    }
                }
                if (index < 0)
                {
                    val.row(i).setZero();
                    continue;
                }
 
                for (int j = 0; j < pts.cols(); ++j)
                    val(i, j) = planar ? initial_acceleration_[index].second[j](pts(i, 0), pts(i, 1)) : initial_acceleration_[index].second[j](pts(i, 0), pts(i, 1), pts(i, 2));
            }
        }
 
        void GenericTensorProblem::clear()
        {
            all_dimensions_dirichlet_ = true;
            has_exact_ = false;
            has_exact_grad_ = false;
            is_time_dept_ = false;
 
            forces_.clear();
            displacements_.clear();
            normal_aligned_forces_.clear();
            pressures_.clear();
            cavity_pressures_.clear();
 
            nodal_dirichlet_.clear();
            nodal_neumann_.clear();
            nodal_dirichlet_mat_.clear();
 
            initial_position_.clear();
            initial_velocity_.clear();
            initial_acceleration_.clear();
 
            for (int i = 0; i < rhs_.size(); ++i)
                rhs_[i].clear();
            for (int i = 0; i < exact_.size(); ++i)
                exact_[i].clear();
            for (int i = 0; i < exact_grad_.size(); ++i)
                exact_grad_[i].clear();
            is_all_ = false;
        }
 
        GenericScalarProblem::GenericScalarProblem(const std::string &name)
            : Problem(name), is_all_(false)
        {
        }
 
        void GenericScalarProblem::set_units(const assembler::Assembler &assembler, const Units &units)
        {
            // TODO?
 
            for (auto &v : neumann_)
                v.set_unit_type("");
 
            for (auto &v : dirichlet_)
                v.set_unit_type("");
 
            for (auto &v : initial_solution_)
                v.second.set_unit_type("");
 
            rhs_.set_unit_type("");
            exact_.set_unit_type("");
 
            for (int i = 0; i < 3; ++i)
                exact_grad_[i].set_unit_type("");
        }
 
        void GenericScalarProblem::rhs(const assembler::Assembler &assembler, const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), 1);
            if (is_rhs_zero())
            {
                val.setZero();
                return;
            }
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                double x = pts(i, 0), y = pts(i, 1), z = planar ? 0 : pts(i, 2);
                val(i) = rhs_(x, y, z, t);
            }
        }
 
        void GenericScalarProblem::dirichlet_bc(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &uv, const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(pts.rows(), 1);
 
            for (long i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_boundary_id(global_ids(i));
                if (is_all_)
                {
                    assert(dirichlet_.size() == 1);
                    val(i) = dirichlet_[0].eval(pts.row(i), t);
                }
                else
                {
                    for (size_t b = 0; b < boundary_ids_.size(); ++b)
                    {
                        if (id == boundary_ids_[b])
                        {
                            val(i) = dirichlet_[b].eval(pts.row(i), t);
                            break;
                        }
                    }
                }
            }
        }
 
        void GenericScalarProblem::neumann_bc(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &uv, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &normals, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(pts.rows(), 1);
 
            for (long i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_boundary_id(global_ids(i));
 
                for (size_t b = 0; b < neumann_boundary_ids_.size(); ++b)
                {
                    if (id == neumann_boundary_ids_[b])
                    {
                        double x = pts(i, 0), y = pts(i, 1), z = pts.cols() == 2 ? 0 : pts(i, 2);
                        val(i) = neumann_[b].eval(pts.row(i), t);
                        break;
                    }
                }
            }
        }
 
        void GenericScalarProblem::initial_solution(const mesh::Mesh &mesh, const Eigen::MatrixXi &global_ids, const Eigen::MatrixXd &pts, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), 1);
            if (initial_solution_.empty())
            {
                val.setZero();
                return;
            }
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                const int id = mesh.get_body_id(global_ids(i));
                int index = -1;
                for (int j = 0; j < initial_solution_.size(); ++j)
                {
                    if (initial_solution_[j].first == id)
                    {
                        index = j;
                        break;
                    }
                }
                if (index < 0)
                {
                    val(i) = 0;
                    continue;
                }
 
                val(i) = planar ? initial_solution_[index].second(pts(i, 0), pts(i, 1)) : initial_solution_[index].second(pts(i, 0), pts(i, 1), pts(i, 2));
            }
        }
 
        void GenericScalarProblem::exact(const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            assert(has_exact_sol());
            const bool planar = pts.cols() == 2;
            val.resize(pts.rows(), 1);
 
            for (int i = 0; i < pts.rows(); ++i)
            {
                double x = pts(i, 0), y = pts(i, 1), z = pts.cols() == 2 ? 0 : pts(i, 2);
                val(i) = exact_(x, y, z, t);
            }
        }
 
        void GenericScalarProblem::exact_grad(const Eigen::MatrixXd &pts, const double t, Eigen::MatrixXd &val) const
        {
            val.resize(pts.rows(), pts.cols());
            if (!has_exact_grad_)
                return;
 
            const bool planar = pts.cols() == 2;
            for (int i = 0; i < pts.rows(); ++i)
            {
                for (int j = 0; j < pts.cols(); ++j)
                {
                    double x = pts(i, 0), y = pts(i, 1), z = pts.cols() == 2 ? 0 : pts(i, 2);
                    val(i, j) = exact_grad_[j](x, y, z, t);
                }
            }
        }
        void GenericScalarProblem::dirichlet_nodal_value(const mesh::Mesh &mesh, const int node_id, const RowVectorNd &pt, const double t, Eigen::MatrixXd &val) const
        {
            val = Eigen::MatrixXd::Zero(1, 1);
            const int tag = mesh.get_node_id(node_id);
 
            if (is_all_)
            {
                assert(nodal_dirichlet_.size() == 1);
                const auto &tmp = nodal_dirichlet_.begin()->second;
 
                val(0) = tmp.eval(pt, t);
                return;
            }
 
            const auto it = nodal_dirichlet_.find(tag);
            if (it != nodal_dirichlet_.end())
            {
                val(0) = it->second.eval(pt, t);
                return;
            }
 
            for (const auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int i = 0; i < n_dirichlet.rows(); ++i)
                {
                    if (n_dirichlet(i, 0) == node_id)
                    {
                        val(0) = n_dirichlet(i, 1);
                        return;
                    }
                }
            }
 
            assert(false);
        }
 
        void GenericScalarProblem::neumann_nodal_value(const mesh::Mesh &mesh, const int node_id, const RowVectorNd &pt, const Eigen::MatrixXd &normal, const double t, Eigen::MatrixXd &val) const
        {
            // TODO implement me;
            log_and_throw_error("Nodal neumann not implemented");
        }
 
        bool GenericScalarProblem::is_nodal_dirichlet_boundary(const int n_id, const int tag)
        {
            if (nodal_dirichlet_.find(tag) != nodal_dirichlet_.end())
                return true;
 
            for (const auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int i = 0; i < n_dirichlet.rows(); ++i)
                {
                    if (n_dirichlet(i, 0) == n_id)
                        return true;
                }
            }
 
            return false;
        }
 
        bool GenericScalarProblem::is_nodal_neumann_boundary(const int n_id, const int tag)
        {
            return nodal_neumann_.find(tag) != nodal_neumann_.end();
        }
 
        bool GenericScalarProblem::has_nodal_dirichlet()
        {
            return nodal_dirichlet_mat_.size() > 0;
        }
 
        bool GenericScalarProblem::has_nodal_neumann()
        {
            return false; // nodal_neumann_.size() > 0;
        }
 
        void GenericScalarProblem::update_nodes(const Eigen::VectorXi &in_node_to_node)
        {
            for (auto &n_dirichlet : nodal_dirichlet_mat_)
            {
                for (int n = 0; n < n_dirichlet.rows(); ++n)
                {
                    const int node_id = in_node_to_node[n_dirichlet(n, 0)];
                    n_dirichlet(n, 0) = node_id;
                }
            }
        }
 
        void GenericScalarProblem::set_parameters(const json &params)
        {
            if (is_param_valid(params, "is_time_dependent"))
            {
                is_time_dept_ = params["is_time_dependent"];
            }
 
            if (is_param_valid(params, "rhs"))
            {
                rhs_.init(params["rhs"]);
            }
 
            if (is_param_valid(params, "reference") && is_param_valid(params["reference"], "solution"))
            {
                has_exact_ = !params["reference"]["solution"].empty();
                exact_.init(params["reference"]["solution"]);
            }
 
            if (is_param_valid(params, "reference") && is_param_valid(params["reference"], "gradient"))
            {
                auto ex = params["reference"]["gradient"];
                has_exact_grad_ = ex.size() > 0;
                if (ex.is_array())
                {
                    for (size_t k = 0; k < ex.size(); ++k)
                        exact_grad_[k].init(ex[k]);
                }
                else
                {
                    assert(false);
                }
            }
 
            if (is_param_valid(params, "dirichlet_boundary"))
            {
                // boundary_ids_.clear();
                const int offset = boundary_ids_.size();
                std::vector<json> j_boundary = flatten_ids(params["dirichlet_boundary"]);
 
                boundary_ids_.resize(offset + j_boundary.size());
                dirichlet_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < boundary_ids_.size(); ++i)
                {
                    if (j_boundary[i - offset].is_string())
                    {
                        const std::string path = resolve_path(j_boundary[i - offset], params["root_path"]);
                        if (!std::filesystem::is_regular_file(path))
                            log_and_throw_error(fmt::format("unable to open {} file", path));
 
                        Eigen::MatrixXd tmp;
                        io::read_matrix(path, tmp);
                        nodal_dirichlet_mat_.emplace_back(tmp);
 
                        continue;
                    }
 
                    int current_id = -1;
 
                    if (j_boundary[i - offset]["id"] == "all")
                    {
                        assert(boundary_ids_.size() == 1);
 
                        is_all_ = true;
                        boundary_ids_.clear();
                        nodal_dirichlet_[current_id] = ScalarBCValue();
                    }
                    else
                    {
                        boundary_ids_[i] = j_boundary[i - offset]["id"];
                        current_id = boundary_ids_[i];
                        nodal_dirichlet_[current_id] = ScalarBCValue();
                    }
 
                    auto ff = j_boundary[i - offset]["value"];
                    dirichlet_[i].value.init(ff);
                    nodal_dirichlet_[current_id].value.init(ff);
 
                    if (j_boundary[i - offset]["interpolation"].is_array())
                    {
                        if (j_boundary[i - offset]["interpolation"].size() == 0)
                            dirichlet_[i].interpolation = std::make_shared<NoInterpolation>();
                        else if (j_boundary[i - offset]["interpolation"].size() == 1)
                            dirichlet_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"][0]);
                        else
                            log_and_throw_error("Only one Dirichlet interpolation supported");
                    }
                    else
                        dirichlet_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"]);
 
                    nodal_dirichlet_[current_id].interpolation = dirichlet_[i].interpolation;
                }
            }
 
            if (is_param_valid(params, "neumann_boundary"))
            {
                // neumann_boundary_ids_.clear();
                const int offset = neumann_boundary_ids_.size();
                auto j_boundary_tmp = params["neumann_boundary"];
                std::vector<json> j_boundary = flatten_ids(j_boundary_tmp);
 
                neumann_boundary_ids_.resize(offset + j_boundary.size());
                neumann_.resize(offset + j_boundary.size());
 
                for (size_t i = offset; i < neumann_boundary_ids_.size(); ++i)
                {
                    neumann_boundary_ids_[i] = j_boundary[i - offset]["id"];
 
                    auto ff = j_boundary[i - offset]["value"];
                    neumann_[i].value.init(ff);
 
                    if (j_boundary[i - offset]["interpolation"].is_array())
                    {
                        if (j_boundary[i - offset]["interpolation"].size() == 0)
                            neumann_[i].interpolation = std::make_shared<NoInterpolation>();
                        else if (j_boundary[i - offset]["interpolation"].size() == 1)
                            neumann_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"][0]);
                        else
                            log_and_throw_error("Only one Neumann interpolation supported");
                    }
                    else
                        neumann_[i].interpolation = Interpolation::build(j_boundary[i - offset]["interpolation"]);
                }
            }
 
            if (is_param_valid(params, "solution"))
            {
                auto rr = params["solution"];
                initial_solution_.resize(rr.size());
                assert(rr.is_array());
 
                for (size_t k = 0; k < rr.size(); ++k)
                {
                    initial_solution_[k].first = rr[k]["id"];
                    initial_solution_[k].second.init(rr[k]["value"]);
                }
            }
        }
 
        void GenericScalarProblem::add_dirichlet_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            boundary_ids_.push_back(id);
            dirichlet_.emplace_back();
            dirichlet_.back().value.init(val);
            dirichlet_.back().interpolation = interp;
        }
 
        void GenericScalarProblem::update_dirichlet_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            dirichlet_[index].value.init(val);
            dirichlet_[index].interpolation = interp;
        }
 
        void GenericScalarProblem::add_neumann_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            neumann_boundary_ids_.push_back(id);
            neumann_.emplace_back();
            neumann_.back().value.init(val);
            neumann_.back().interpolation = interp;
        }
 
        void GenericScalarProblem::update_neumann_boundary(const int id, const double val, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            neumann_[index].value.init(val);
            neumann_[index].interpolation = interp;
        }
 
        void GenericScalarProblem::add_dirichlet_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            boundary_ids_.push_back(id);
            dirichlet_.emplace_back();
            dirichlet_.back().value.init(func);
            dirichlet_.back().interpolation = interp;
        }
 
        void GenericScalarProblem::update_dirichlet_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            dirichlet_[index].value.init(func);
            dirichlet_[index].interpolation = interp;
        }
 
        void GenericScalarProblem::add_neumann_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            neumann_boundary_ids_.push_back(id);
            neumann_.emplace_back();
            neumann_.back().value.init(func);
            neumann_.back().interpolation = interp;
        }
 
        void GenericScalarProblem::update_neumann_boundary(const int id, const std::function<double(double x, double y, double z, double t)> &func, const std::shared_ptr<Interpolation> &interp)
        {
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            neumann_[index].value.init(func);
            neumann_[index].interpolation = interp;
        }
 
        void GenericScalarProblem::add_dirichlet_boundary(const int id, const json &val, const std::string &interp)
        {
            boundary_ids_.push_back(id);
            dirichlet_.emplace_back();
            dirichlet_.back().value.init(val);
            if (interp.empty())
                dirichlet_.back().interpolation = std::make_shared<NoInterpolation>();
            else
                dirichlet_.back().interpolation = Interpolation::build(interp);
        }
 
        void GenericScalarProblem::add_neumann_boundary(const int id, const json &val, const std::string &interp)
        {
            neumann_boundary_ids_.push_back(id);
            neumann_.emplace_back();
            neumann_.back().value.init(val);
            if (interp.empty())
                neumann_.back().interpolation = std::make_shared<NoInterpolation>();
            else
                neumann_.back().interpolation = Interpolation::build(interp);
        }
 
        void GenericScalarProblem::update_dirichlet_boundary(const int id, const json &val, const std::string &interp)
        {
            int index = -1;
            for (int i = 0; i < boundary_ids_.size(); ++i)
            {
                if (boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
 
            dirichlet_[index].value.init(val);
            if (interp.empty())
                dirichlet_[index].interpolation = std::make_shared<NoInterpolation>();
            else
                dirichlet_[index].interpolation = Interpolation::build(interp);
        }
 
        void GenericScalarProblem::update_neumann_boundary(const int id, const json &val, const std::string &interp)
        {
            int index = -1;
            for (int i = 0; i < neumann_boundary_ids_.size(); ++i)
            {
                if (neumann_boundary_ids_[i] == id)
                {
                    index = i;
                    break;
                }
            }
            if (index == -1)
            {
                throw "Invalid boundary id";
            }
            neumann_[index].value.init(val);
 
            if (interp.empty())
                neumann_[index].interpolation = std::make_shared<NoInterpolation>();
            else
                neumann_[index].interpolation = Interpolation::build(interp);
        }
 
        void GenericScalarProblem::clear()
        {
            neumann_.clear();
            dirichlet_.clear();
 
            nodal_dirichlet_.clear();
            nodal_neumann_.clear();
            nodal_dirichlet_mat_.clear();
 
            rhs_.clear();
            exact_.clear();
            for (int i = 0; i < exact_grad_.size(); ++i)
                exact_grad_[i].clear();
            is_all_ = false;
            has_exact_ = false;
            has_exact_grad_ = false;
            is_time_dept_ = false;
        }
    } // namespace assembler
} // namespace polyfem