State.cpp

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#include <polyfem/State.hpp>
#include <polyfem/Common.hpp>
 
#include <polyfem/io/MatrixIO.hpp>
#include <polyfem/io/Evaluator.hpp>
 
#include <polyfem/assembler/Mass.hpp>
#include <polyfem/assembler/MultiModel.hpp>
 
#include <polyfem/mesh/mesh2D/Mesh2D.hpp>
#include <polyfem/mesh/mesh2D/CMesh2D.hpp>
#include <polyfem/mesh/mesh2D/NCMesh2D.hpp>
#include <polyfem/mesh/mesh3D/Mesh3D.hpp>
#include <polyfem/mesh/mesh3D/CMesh3D.hpp>
#include <polyfem/mesh/mesh3D/NCMesh3D.hpp>
#include <polyfem/mesh/MeshUtils.hpp>
#include <polyfem/mesh/collision_proxy/CollisionProxy.hpp>
 
#include <polyfem/basis/LagrangeBasis2d.hpp>
#include <polyfem/basis/LagrangeBasis3d.hpp>
 
#include <polyfem/refinement/APriori.hpp>
 
#include <polyfem/basis/SplineBasis2d.hpp>
#include <polyfem/basis/SplineBasis3d.hpp>
 
#include <polyfem/basis/barycentric/MVPolygonalBasis2d.hpp>
#include <polyfem/basis/barycentric/WSPolygonalBasis2d.hpp>
 
#include <polyfem/basis/PolygonalBasis2d.hpp>
#include <polyfem/basis/PolygonalBasis3d.hpp>
 
#include <polyfem/autogen/auto_p_bases.hpp>
#include <polyfem/autogen/auto_q_bases.hpp>
 
#include <polyfem/quadrature/HexQuadrature.hpp>
#include <polyfem/quadrature/QuadQuadrature.hpp>
#include <polyfem/quadrature/TetQuadrature.hpp>
#include <polyfem/quadrature/TriQuadrature.hpp>
 
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/Timer.hpp>
 
#include <polysolve/linear/FEMSolver.hpp>
 
#include <polyfem/io/OBJWriter.hpp>
 
#include <igl/edges.h>
#include <igl/Timer.h>
 
#include <iostream>
#include <algorithm>
#include <memory>
#include <filesystem>
 
#include <polyfem/io/Evaluator.hpp>
 
#include <polyfem/utils/autodiff.h>
DECLARE_DIFFSCALAR_BASE();
 
using namespace Eigen;
 
namespace polyfem
{
    using namespace assembler;
    using namespace mesh;
    using namespace io;
    using namespace utils;
 
    namespace
    {
        void build_in_node_to_in_primitive(const Mesh &mesh, const MeshNodes &mesh_nodes,
                                           Eigen::VectorXi &in_node_to_in_primitive,
                                           Eigen::VectorXi &in_node_offset)
        {
            const int num_vertex_nodes = mesh_nodes.num_vertex_nodes();
            const int num_edge_nodes = mesh_nodes.num_edge_nodes();
            const int num_face_nodes = mesh_nodes.num_face_nodes();
            const int num_cell_nodes = mesh_nodes.num_cell_nodes();
 
            const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
 
            const long n_vertices = num_vertex_nodes;
            const int num_in_primitives = n_vertices + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
            const int num_primitives = mesh.n_vertices() + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
 
            in_node_to_in_primitive.resize(num_nodes);
            in_node_offset.resize(num_nodes);
 
            // Only one node per vertex, so this is an identity map.
            in_node_to_in_primitive.head(num_vertex_nodes).setLinSpaced(num_vertex_nodes, 0, num_vertex_nodes - 1); // vertex nodes
            in_node_offset.head(num_vertex_nodes).setZero();
 
            int prim_offset = n_vertices;
            int node_offset = num_vertex_nodes;
            auto foo = [&](const int num_prims, const int num_prim_nodes) {
                if (num_prims <= 0 || num_prim_nodes <= 0)
                    return;
                const Eigen::VectorXi range = Eigen::VectorXi::LinSpaced(num_prim_nodes, 0, num_prim_nodes - 1);
                // TODO: This assumes isotropic degree of element.
                const int node_per_prim = num_prim_nodes / num_prims;
 
                in_node_to_in_primitive.segment(node_offset, num_prim_nodes) =
                    range.array() / node_per_prim + prim_offset;
 
                in_node_offset.segment(node_offset, num_prim_nodes) =
                    range.unaryExpr([&](const int x) { return x % node_per_prim; });
 
                prim_offset += num_prims;
                node_offset += num_prim_nodes;
            };
 
            foo(mesh.n_edges(), num_edge_nodes);
            foo(mesh.n_faces(), num_face_nodes);
            foo(mesh.n_cells(), num_cell_nodes);
        }
 
        bool build_in_primitive_to_primitive(
            const Mesh &mesh, const MeshNodes &mesh_nodes,
            const Eigen::VectorXi &in_ordered_vertices,
            const Eigen::MatrixXi &in_ordered_edges,
            const Eigen::MatrixXi &in_ordered_faces,
            Eigen::VectorXi &in_primitive_to_primitive)
        {
            // NOTE: Assume in_cells_to_cells is identity
            const int num_vertex_nodes = mesh_nodes.num_vertex_nodes();
            const int num_edge_nodes = mesh_nodes.num_edge_nodes();
            const int num_face_nodes = mesh_nodes.num_face_nodes();
            const int num_cell_nodes = mesh_nodes.num_cell_nodes();
            const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
 
            const long n_vertices = num_vertex_nodes;
            const int num_in_primitives = n_vertices + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
            const int num_primitives = mesh.n_vertices() + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
 
            in_primitive_to_primitive.setLinSpaced(num_in_primitives, 0, num_in_primitives - 1);
 
            igl::Timer timer;
 
            // ------------
            // Map vertices
            // ------------
 
            if (in_ordered_vertices.rows() != n_vertices)
            {
                logger().warn("Node ordering disabled, in_ordered_vertices != n_vertices, {} != {}", in_ordered_vertices.rows(), n_vertices);
                return false;
            }
 
            in_primitive_to_primitive.head(n_vertices) = in_ordered_vertices;
 
            int in_offset = n_vertices;
            int offset = mesh.n_vertices();
 
            // ---------
            // Map edges
            // ---------
 
            logger().trace("Building Mesh edges to IDs...");
            timer.start();
            const auto edges_to_ids = mesh.edges_to_ids();
            if (in_ordered_edges.rows() != edges_to_ids.size())
            {
                logger().warn("Node ordering disabled, in_ordered_edges != edges_to_ids, {} != {}", in_ordered_edges.rows(), edges_to_ids.size());
                return false;
            }
            timer.stop();
            logger().trace("Done (took {}s)", timer.getElapsedTime());
 
            logger().trace("Building in-edge to edge mapping...");
            timer.start();
            for (int in_ei = 0; in_ei < in_ordered_edges.rows(); in_ei++)
            {
                const std::pair<int, int> in_edge(
                    in_ordered_edges.row(in_ei).minCoeff(),
                    in_ordered_edges.row(in_ei).maxCoeff());
                in_primitive_to_primitive[in_offset + in_ei] =
                    offset + edges_to_ids.at(in_edge); // offset edge ids
            }
            timer.stop();
            logger().trace("Done (took {}s)", timer.getElapsedTime());
 
            in_offset += mesh.n_edges();
            offset += mesh.n_edges();
 
            // ---------
            // Map faces
            // ---------
 
            if (mesh.is_volume())
            {
                logger().trace("Building Mesh faces to IDs...");
                timer.start();
                const auto faces_to_ids = mesh.faces_to_ids();
                if (in_ordered_faces.rows() != faces_to_ids.size())
                {
                    logger().warn("Node ordering disabled, in_ordered_faces != faces_to_ids, {} != {}", in_ordered_faces.rows(), faces_to_ids.size());
                    return false;
                }
                timer.stop();
                logger().trace("Done (took {}s)", timer.getElapsedTime());
 
                logger().trace("Building in-face to face mapping...");
                timer.start();
                for (int in_fi = 0; in_fi < in_ordered_faces.rows(); in_fi++)
                {
                    std::vector<int> in_face(in_ordered_faces.cols());
                    for (int i = 0; i < in_face.size(); i++)
                        in_face[i] = in_ordered_faces(in_fi, i);
                    std::sort(in_face.begin(), in_face.end());
 
                    in_primitive_to_primitive[in_offset + in_fi] =
                        offset + faces_to_ids.at(in_face); // offset face ids
                }
                timer.stop();
                logger().trace("Done (took {}s)", timer.getElapsedTime());
 
                in_offset += mesh.n_faces();
                offset += mesh.n_faces();
            }
 
            return true;
        }
    } // namespace
 
    std::vector<int> State::primitive_to_node() const
    {
        auto indices = iso_parametric() ? mesh_nodes->primitive_to_node() : geom_mesh_nodes->primitive_to_node();
        indices.resize(mesh->n_vertices());
        return indices;
    }
 
    std::vector<int> State::node_to_primitive() const
    {
        auto p2n = primitive_to_node();
        std::vector<int> indices;
        indices.resize(n_geom_bases);
        for (int i = 0; i < p2n.size(); i++)
            indices[p2n[i]] = i;
        return indices;
    }
 
    void State::build_node_mapping()
    {
        if (args["space"]["basis_type"] == "Spline")
        {
            logger().warn("Node ordering disabled, it dosent work for splines!");
            return;
        }
 
        if (disc_orders.maxCoeff() >= 4 || disc_orders.maxCoeff() != disc_orders.minCoeff())
        {
            logger().warn("Node ordering disabled, it works only for p < 4 and uniform order!");
            return;
        }
 
        if (!mesh->is_conforming())
        {
            logger().warn("Node ordering disabled, not supported for non-conforming meshes!");
            return;
        }
 
        if (mesh->has_poly())
        {
            logger().warn("Node ordering disabled, not supported for polygonal meshes!");
            return;
        }
 
        if (mesh->in_ordered_vertices().size() <= 0 || mesh->in_ordered_edges().size() <= 0 || (mesh->is_volume() && mesh->in_ordered_faces().size() <= 0))
        {
            logger().warn("Node ordering disabled, input vertices/edges/faces not computed!");
            return;
        }
 
        const int num_vertex_nodes = mesh_nodes->num_vertex_nodes();
        const int num_edge_nodes = mesh_nodes->num_edge_nodes();
        const int num_face_nodes = mesh_nodes->num_face_nodes();
        const int num_cell_nodes = mesh_nodes->num_cell_nodes();
 
        const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
 
        const long n_vertices = num_vertex_nodes;
        const int num_in_primitives = n_vertices + mesh->n_edges() + mesh->n_faces() + mesh->n_cells();
        const int num_primitives = mesh->n_vertices() + mesh->n_edges() + mesh->n_faces() + mesh->n_cells();
 
        igl::Timer timer;
 
        logger().trace("Building in-node to in-primitive mapping...");
        timer.start();
        Eigen::VectorXi in_node_to_in_primitive;
        Eigen::VectorXi in_node_offset;
        build_in_node_to_in_primitive(*mesh, *mesh_nodes, in_node_to_in_primitive, in_node_offset);
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
 
        logger().trace("Building in-primitive to primitive mapping...");
        timer.start();
        bool ok = build_in_primitive_to_primitive(
            *mesh, *mesh_nodes,
            mesh->in_ordered_vertices(),
            mesh->in_ordered_edges(),
            mesh->in_ordered_faces(),
            in_primitive_to_primitive);
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
 
        if (!ok)
        {
            in_node_to_node.resize(0);
            in_primitive_to_primitive.resize(0);
            return;
        }
 
        const auto primitive_offset = [&](int node) {
            if (mesh_nodes->is_vertex_node(node))
                return 0;
            else if (mesh_nodes->is_edge_node(node))
                return mesh->n_vertices();
            else if (mesh_nodes->is_face_node(node))
                return mesh->n_vertices() + mesh->n_edges();
            else if (mesh_nodes->is_cell_node(node))
                return mesh->n_vertices() + mesh->n_edges() + mesh->n_faces();
            throw std::runtime_error("Invalid node ID!");
        };
 
        logger().trace("Building primitive to node mapping...");
        timer.start();
        std::vector<std::vector<int>> primitive_to_nodes(num_primitives);
        const std::vector<int> &grouped_nodes = mesh_nodes->primitive_to_node();
        int node_count = 0;
        for (int i = 0; i < grouped_nodes.size(); i++)
        {
            int node = grouped_nodes[i];
            assert(node < num_nodes);
            if (node >= 0)
            {
                int primitive = mesh_nodes->node_to_primitive_gid().at(node) + primitive_offset(i);
                assert(primitive < num_primitives);
                primitive_to_nodes[primitive].push_back(node);
                node_count++;
            }
        }
        assert(node_count == num_nodes);
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
 
        logger().trace("Combining mappings...");
        timer.start();
        in_node_to_node.setConstant(num_nodes, -1);
        for (int i = 0; i < num_nodes; i++)
        {
            // input node id -> input primitive -> primitive -> node(s)
            const std::vector<int> &possible_nodes =
                primitive_to_nodes[in_primitive_to_primitive[in_node_to_in_primitive[i]]];
 
            if (possible_nodes.size() == 1)
                in_node_to_node[i] = possible_nodes[0];
            else
            {
                assert(possible_nodes.size() > 1);
 
                // TODO: The following code assumes multiple nodes must come from an edge.
                //       This only true for P3. P4+ has multiple face nodes and P5+ have multiple cell nodes.
 
                int e_id = in_primitive_to_primitive[in_node_to_in_primitive[i]] - mesh->n_vertices();
                assert(e_id < mesh->n_edges());
                assert(in_node_to_node[mesh->edge_vertex(e_id, 0)] >= 0); // Vertex nodes should be computed first
                RowVectorNd v0 = mesh_nodes->node_position(in_node_to_node[mesh->edge_vertex(e_id, 0)]);
                RowVectorNd a = mesh_nodes->node_position(possible_nodes[0]);
                RowVectorNd b = mesh_nodes->node_position(possible_nodes[1]);
                // Assume possible nodes are ordered, so only need to check order of two nodes
 
                // Input edges are sorted, so if a is closer to v0 then the order is correct
                // otherwise the nodes are flipped.
                assert(mesh->edge_vertex(e_id, 0) < mesh->edge_vertex(e_id, 1));
                int offset = (a - v0).squaredNorm() < (b - v0).squaredNorm()
                                 ? in_node_offset[i]
                                 : (possible_nodes.size() - in_node_offset[i] - 1);
                in_node_to_node[i] = possible_nodes[offset];
            }
        }
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
    }
 
    std::string State::formulation() const
    {
        if (args["materials"].is_null())
        {
            logger().error("specify some 'materials'");
            assert(!args["materials"].is_null());
            throw "invalid input";
        }
 
        if (args["materials"].is_array())
        {
            std::string current = "";
            for (const auto &m : args["materials"])
            {
                const std::string tmp = m["type"];
                if (current.empty())
                    current = tmp;
                else if (current != tmp)
                {
                    if (current == "LinearElasticity"
                        || current == "NeoHookean"
                        || current == "SaintVenant"
                        || current == "HookeLinearElasticity"
                        || current == "MooneyRivlin"
                        || current == "MooneyRivlin3Param"
                        || current == "UnconstrainedOgden"
                        || current == "IncompressibleOgden"
                        || current == "MultiModels")
                    {
                        if (tmp == "LinearElasticity"
                            || tmp == "NeoHookean"
                            || tmp == "SaintVenant"
                            || tmp == "HookeLinearElasticity"
                            || tmp == "MooneyRivlin"
                            || current == "MooneyRivlin3Param"
                            || tmp == "UnconstrainedOgden"
                            || tmp == "IncompressibleOgden")
                            current = "MultiModels";
                        else
                        {
                            logger().error("Current material is {}, new material is {}, multimaterial supported only for LinearElasticity and NeoHookean", current, tmp);
                            throw "invalid input";
                        }
                    }
                    else
                    {
                        logger().error("Current material is {}, new material is {}, multimaterial supported only for LinearElasticity and NeoHookean", current, tmp);
                        throw "invalid input";
                    }
                }
            }
 
            return current;
        }
        else
            return args["materials"]["type"];
    }
 
    void State::sol_to_pressure(Eigen::MatrixXd &sol, Eigen::MatrixXd &pressure)
    {
        if (n_pressure_bases <= 0)
        {
            logger().error("No pressure bases defined!");
            return;
        }
 
        assert(mixed_assembler != nullptr);
        Eigen::MatrixXd tmp = sol;
 
        int fluid_offset = use_avg_pressure ? (assembler->is_fluid() ? 1 : 0) : 0;
        sol = tmp.topRows(tmp.rows() - n_pressure_bases - fluid_offset);
        assert(sol.size() == n_bases * (problem->is_scalar() ? 1 : mesh->dimension()));
        pressure = tmp.middleRows(tmp.rows() - n_pressure_bases - fluid_offset, n_pressure_bases);
        assert(pressure.size() == n_pressure_bases);
    }
 
    void compute_integral_constraints(
        const Mesh3D &mesh,
        const int n_bases,
        const std::vector<basis::ElementBases> &bases,
        const std::vector<basis::ElementBases> &gbases,
        Eigen::MatrixXd &basis_integrals)
    {
        if (!mesh.is_volume())
        {
            logger().error("Works only on volumetric meshes!");
            return;
        }
        assert(mesh.is_volume());
 
        basis_integrals.resize(n_bases, 9);
        basis_integrals.setZero();
        Eigen::MatrixXd rhs(n_bases, 9);
        rhs.setZero();
 
        const int n_elements = mesh.n_elements();
        for (int e = 0; e < n_elements; ++e)
        {
            // if (mesh.is_polytope(e)) {
            //  continue;
            // }
            // ElementAssemblyValues vals = values[e];
            // const ElementAssemblyValues &gvals = gvalues[e];
            ElementAssemblyValues vals;
            vals.compute(e, mesh.is_volume(), bases[e], gbases[e]);
 
            // Computes the discretized integral of the PDE over the element
            const int n_local_bases = int(vals.basis_values.size());
            for (int j = 0; j < n_local_bases; ++j)
            {
                const AssemblyValues &v = vals.basis_values[j];
                const double integral_100 = (v.grad_t_m.col(0).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_010 = (v.grad_t_m.col(1).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_001 = (v.grad_t_m.col(2).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
 
                const double integral_110 = ((vals.val.col(1).array() * v.grad_t_m.col(0).array() + vals.val.col(0).array() * v.grad_t_m.col(1).array()) * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_011 = ((vals.val.col(2).array() * v.grad_t_m.col(1).array() + vals.val.col(1).array() * v.grad_t_m.col(2).array()) * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_101 = ((vals.val.col(0).array() * v.grad_t_m.col(2).array() + vals.val.col(2).array() * v.grad_t_m.col(0).array()) * vals.det.array() * vals.quadrature.weights.array()).sum();
 
                const double integral_200 = 2 * (vals.val.col(0).array() * v.grad_t_m.col(0).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_020 = 2 * (vals.val.col(1).array() * v.grad_t_m.col(1).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
                const double integral_002 = 2 * (vals.val.col(2).array() * v.grad_t_m.col(2).array() * vals.det.array() * vals.quadrature.weights.array()).sum();
 
                const double area = (v.val.array() * vals.det.array() * vals.quadrature.weights.array()).sum();
 
                for (size_t ii = 0; ii < v.global.size(); ++ii)
                {
                    basis_integrals(v.global[ii].index, 0) += integral_100 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 1) += integral_010 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 2) += integral_001 * v.global[ii].val;
 
                    basis_integrals(v.global[ii].index, 3) += integral_110 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 4) += integral_011 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 5) += integral_101 * v.global[ii].val;
 
                    basis_integrals(v.global[ii].index, 6) += integral_200 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 7) += integral_020 * v.global[ii].val;
                    basis_integrals(v.global[ii].index, 8) += integral_002 * v.global[ii].val;
 
                    rhs(v.global[ii].index, 6) += -2.0 * area * v.global[ii].val;
                    rhs(v.global[ii].index, 7) += -2.0 * area * v.global[ii].val;
                    rhs(v.global[ii].index, 8) += -2.0 * area * v.global[ii].val;
                }
            }
        }
 
        basis_integrals -= rhs;
    }
 
    bool State::iso_parametric() const
    {
        if (mesh->has_poly())
            return true;
 
        if (args["space"]["basis_type"] == "Spline")
            return true;
 
        if (mesh->is_rational())
            return false;
 
        if (args["space"]["use_p_ref"])
            return false;
 
        if (has_periodic_bc())
            return false;
 
        if (optimization_enabled == solver::CacheLevel::Derivatives)
            return false;
 
        if (mesh->orders().size() <= 0)
        {
            if (args["space"]["discr_order"] == 1)
                return true;
            else
                return args["space"]["advanced"]["isoparametric"];
        }
 
        if (mesh->orders().minCoeff() != mesh->orders().maxCoeff())
            return false;
 
        if (args["space"]["discr_order"] == mesh->orders().minCoeff())
            return true;
 
        // TODO:
        // if (args["space"]["discr_order"] == 1 && args["force_linear_geometry"])
        //  return true;
 
        return args["space"]["advanced"]["isoparametric"];
    }
 
    void State::build_basis()
    {
        if (!mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
 
        mesh->prepare_mesh();
 
        bases.clear();
        pressure_bases.clear();
        geom_bases_.clear();
        boundary_nodes.clear();
        dirichlet_nodes.clear();
        neumann_nodes.clear();
        local_boundary.clear();
        total_local_boundary.clear();
        local_neumann_boundary.clear();
        local_pressure_boundary.clear();
        local_pressure_cavity.clear();
        polys.clear();
        poly_edge_to_data.clear();
        rhs.resize(0, 0);
 
        if (assembler::MultiModel *mm = dynamic_cast<assembler::MultiModel *>(assembler.get()))
        {
            assert(args["materials"].is_array());
 
            std::vector<std::string> materials(mesh->n_elements());
 
            std::map<int, std::string> mats;
 
            for (const auto &m : args["materials"])
                mats[m["id"].get<int>()] = m["type"];
 
            for (int i = 0; i < materials.size(); ++i)
                materials[i] = mats.at(mesh->get_body_id(i));
 
            mm->init_multimodels(materials);
        }
 
        n_bases = 0;
        n_geom_bases = 0;
        n_pressure_bases = 0;
 
        stats.reset();
 
        disc_orders.resize(mesh->n_elements());
 
        problem->init(*mesh);
        logger().info("Building {} basis...", (iso_parametric() ? "isoparametric" : "not isoparametric"));
        const bool has_polys = mesh->has_poly();
 
        local_boundary.clear();
        local_neumann_boundary.clear();
        local_pressure_boundary.clear();
        local_pressure_cavity.clear();
        std::map<int, basis::InterfaceData> poly_edge_to_data_geom; // temp dummy variable
 
        const auto &tmp_json = args["space"]["discr_order"];
        if (tmp_json.is_number_integer())
        {
            disc_orders.setConstant(tmp_json);
        }
        else if (tmp_json.is_string())
        {
            const std::string discr_orders_path = tmp_json;
            Eigen::MatrixXi tmp;
            read_matrix(discr_orders_path, tmp);
            assert(tmp.size() == disc_orders.size());
            assert(tmp.cols() == 1);
            disc_orders = tmp;
        }
        else if (tmp_json.is_array())
        {
            const auto b_discr_orders = tmp_json;
 
            std::map<int, int> b_orders;
            for (size_t i = 0; i < b_discr_orders.size(); ++i)
            {
                assert(b_discr_orders[i]["id"].is_array() || b_discr_orders[i]["id"].is_number_integer());
 
                const int order = b_discr_orders[i]["order"];
                for (const int id : json_as_array<int>(b_discr_orders[i]["id"]))
                {
                    b_orders[id] = order;
                    logger().trace("bid {}, discr {}", id, order);
                }
            }
 
            for (int e = 0; e < mesh->n_elements(); ++e)
            {
                const int bid = mesh->get_body_id(e);
                const auto order = b_orders.find(bid);
                if (order == b_orders.end())
                {
                    logger().debug("Missing discretization order for body {}; using 1", bid);
                    b_orders[bid] = 1;
                    disc_orders[e] = 1;
                }
                else
                {
                    disc_orders[e] = order->second;
                }
            }
        }
        else
        {
            logger().error("space/discr_order must be either a number a path or an array");
            throw "invalid json";
        }
        // TODO: same for pressure!
 
        Eigen::MatrixXi geom_disc_orders;
        if (!iso_parametric())
        {
            if (mesh->orders().size() <= 0)
            {
                geom_disc_orders.resizeLike(disc_orders);
                geom_disc_orders.setConstant(1);
            }
            else
                geom_disc_orders = mesh->orders();
        }
 
        igl::Timer timer;
        timer.start();
        if (args["space"]["use_p_ref"])
        {
            refinement::APriori::p_refine(
                *mesh,
                args["space"]["advanced"]["B"],
                args["space"]["advanced"]["h1_formula"],
                args["space"]["discr_order"],
                args["space"]["advanced"]["discr_order_max"],
                stats,
                disc_orders);
 
            logger().info("min p: {} max p: {}", disc_orders.minCoeff(), disc_orders.maxCoeff());
        }
 
        int quadrature_order = args["space"]["advanced"]["quadrature_order"].get<int>();
        const int mass_quadrature_order = args["space"]["advanced"]["mass_quadrature_order"].get<int>();
        if (mixed_assembler != nullptr)
        {
            const int disc_order = disc_orders.maxCoeff();
            if (disc_order - disc_orders.minCoeff() != 0)
            {
                logger().error("p refinement not supported in mixed formulation!");
                return;
            }
        }
 
        // shape optimization needs continuous geometric basis
        // const bool use_continuous_gbasis = optimization_enabled == solver::CacheLevel::Derivatives;
        const bool use_continuous_gbasis = true;
 
        if (mesh->is_volume())
        {
            const Mesh3D &tmp_mesh = *dynamic_cast<Mesh3D *>(mesh.get());
            if (args["space"]["basis_type"] == "Spline")
            {
                // if (!iso_parametric())
                // {
                //  logger().error("Splines must be isoparametric, ignoring...");
                //  // LagrangeBasis3d::build_bases(tmp_mesh, quadrature_order, geom_disc_orders, has_polys, geom_bases_, local_boundary, poly_edge_to_data_geom, mesh_nodes);
                //  SplineBasis3d::build_bases(tmp_mesh, quadrature_order, geom_bases_, local_boundary, poly_edge_to_data);
                // }
 
                n_bases = basis::SplineBasis3d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, bases, local_boundary, poly_edge_to_data);
 
                // if (iso_parametric() && args["fit_nodes"])
                //  SplineBasis3d::fit_nodes(tmp_mesh, n_bases, bases);
            }
            else
            {
                if (!iso_parametric())
                    n_geom_bases = basis::LagrangeBasis3d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, geom_disc_orders, false, has_polys, !use_continuous_gbasis, geom_bases_, local_boundary, poly_edge_to_data_geom, geom_mesh_nodes);
 
                n_bases = basis::LagrangeBasis3d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, disc_orders, args["space"]["basis_type"] == "Serendipity", has_polys, false, bases, local_boundary, poly_edge_to_data, mesh_nodes);
            }
 
            // if(problem->is_mixed())
            if (mixed_assembler != nullptr)
            {
                n_pressure_bases = basis::LagrangeBasis3d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, int(args["space"]["pressure_discr_order"]), false, has_polys, false, pressure_bases, local_boundary, poly_edge_to_data_geom, pressure_mesh_nodes);
            }
        }
        else
        {
            const Mesh2D &tmp_mesh = *dynamic_cast<Mesh2D *>(mesh.get());
            if (args["space"]["basis_type"] == "Spline")
            {
                // TODO:
                // if (!iso_parametric())
                // {
                //  logger().error("Splines must be isoparametric, ignoring...");
                //  // LagrangeBasis2d::build_bases(tmp_mesh, quadrature_order, disc_orders, has_polys, geom_bases_, local_boundary, poly_edge_to_data_geom, mesh_nodes);
                //  n_bases = SplineBasis2d::build_bases(tmp_mesh, quadrature_order, geom_bases_, local_boundary, poly_edge_to_data);
                // }
 
                n_bases = basis::SplineBasis2d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, bases, local_boundary, poly_edge_to_data);
 
                // if (iso_parametric() && args["fit_nodes"])
                //  SplineBasis2d::fit_nodes(tmp_mesh, n_bases, bases);
            }
            else
            {
                if (!iso_parametric())
                    n_geom_bases = basis::LagrangeBasis2d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, geom_disc_orders, false, has_polys, !use_continuous_gbasis, geom_bases_, local_boundary, poly_edge_to_data_geom, geom_mesh_nodes);
 
                n_bases = basis::LagrangeBasis2d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, disc_orders, args["space"]["basis_type"] == "Serendipity", has_polys, false, bases, local_boundary, poly_edge_to_data, mesh_nodes);
            }
 
            // if(problem->is_mixed())
            if (mixed_assembler != nullptr)
            {
                n_pressure_bases = basis::LagrangeBasis2d::build_bases(tmp_mesh, assembler->name(), quadrature_order, mass_quadrature_order, int(args["space"]["pressure_discr_order"]), false, has_polys, false, pressure_bases, local_boundary, poly_edge_to_data_geom, pressure_mesh_nodes);
            }
        }
 
        if (mixed_assembler != nullptr)
        {
            assert(bases.size() == pressure_bases.size());
            for (int i = 0; i < pressure_bases.size(); ++i)
            {
                quadrature::Quadrature b_quad;
                bases[i].compute_quadrature(b_quad);
                pressure_bases[i].set_quadrature([b_quad](quadrature::Quadrature &quad) { quad = b_quad; });
            }
        }
 
        timer.stop();
 
        build_polygonal_basis();
 
        if (n_geom_bases == 0)
            n_geom_bases = n_bases;
 
        auto &gbases = geom_bases();
 
        if (optimization_enabled == solver::CacheLevel::Derivatives)
        {
            std::map<std::array<int, 2>, double> pairs;
            for (int e = 0; e < gbases.size(); e++)
            {
                const auto &gbs = gbases[e].bases;
                const auto &bs = bases[e].bases;
 
                Eigen::MatrixXd local_pts;
                const int order = bs.front().order();
                if (mesh->is_volume())
                {
                    if (mesh->is_simplex(e))
                        autogen::p_nodes_3d(order, local_pts);
                    else
                        autogen::q_nodes_3d(order, local_pts);
                }
                else
                {
                    if (mesh->is_simplex(e))
                        autogen::p_nodes_2d(order, local_pts);
                    else
                        autogen::q_nodes_2d(order, local_pts);
                }
 
                ElementAssemblyValues vals;
                vals.compute(e, mesh->is_volume(), local_pts, gbases[e], gbases[e]);
 
                for (int i = 0; i < bs.size(); i++)
                {
                    for (int j = 0; j < gbs.size(); j++)
                    {
                        if (std::abs(vals.basis_values[j].val(i)) > 1e-7)
                        {
                            std::array<int, 2> index = {{gbs[j].global()[0].index, bs[i].global()[0].index}};
                            pairs.insert({index, vals.basis_values[j].val(i)});
                        }
                    }
                }
            }
 
            const int dim = mesh->dimension();
            std::vector<Eigen::Triplet<double>> coeffs;
            coeffs.clear();
            for (const auto &iter : pairs)
                for (int d = 0; d < dim; d++)
                    coeffs.emplace_back(iter.first[0] * dim + d, iter.first[1] * dim + d, iter.second);
 
            gbasis_nodes_to_basis_nodes.resize(n_geom_bases * mesh->dimension(), n_bases * mesh->dimension());
            gbasis_nodes_to_basis_nodes.setFromTriplets(coeffs.begin(), coeffs.end());
        }
 
        for (const auto &lb : local_boundary)
            total_local_boundary.emplace_back(lb);
 
        const int dim = mesh->dimension();
        const int problem_dim = problem->is_scalar() ? 1 : dim;
 
        // handle periodic bc
        if (has_periodic_bc())
        {
            // collect periodic directions
            json directions = args["boundary_conditions"]["periodic_boundary"]["correspondence"];
            Eigen::MatrixXd tile_offset = Eigen::MatrixXd::Identity(dim, dim);
 
            if (directions.size() > 0)
            {
                Eigen::VectorXd tmp;
                for (int d = 0; d < dim; d++)
                {
                    tmp = directions[d];
                    if (tmp.size() != dim)
                        log_and_throw_error("Invalid size of periodic directions!");
                    tile_offset.col(d) = tmp;
                }
            }
 
            periodic_bc = std::make_shared<PeriodicBoundary>(problem->is_scalar(), n_bases, bases, mesh_nodes, tile_offset, args["boundary_conditions"]["periodic_boundary"]["tolerance"].get<double>());
 
            macro_strain_constraint.init(dim, args["boundary_conditions"]["periodic_boundary"]);
        }
 
        if (args["space"]["advanced"]["count_flipped_els"])
            stats.count_flipped_elements(*mesh, geom_bases());
 
        const int prev_bases = n_bases;
        n_bases += obstacle.n_vertices();
 
        {
            igl::Timer timer2;
            logger().debug("Building node mapping...");
            timer2.start();
            build_node_mapping();
            problem->update_nodes(in_node_to_node);
            mesh->update_nodes(in_node_to_node);
            timer2.stop();
            logger().debug("Done (took {}s)", timer2.getElapsedTime());
        }
 
        logger().info("Building collision mesh...");
        build_collision_mesh();
        if (periodic_bc && args["contact"]["periodic"])
            build_periodic_collision_mesh();
        logger().info("Done!");
 
        const int prev_b_size = local_boundary.size();
        problem->setup_bc(*mesh, n_bases - obstacle.n_vertices(),
                          bases, geom_bases(), pressure_bases,
                          local_boundary,
                          boundary_nodes,
                          local_neumann_boundary,
                          local_pressure_boundary,
                          local_pressure_cavity,
                          pressure_boundary_nodes,
                          dirichlet_nodes, neumann_nodes);
 
        // setp nodal values
        {
            dirichlet_nodes_position.resize(dirichlet_nodes.size());
            for (int n = 0; n < dirichlet_nodes.size(); ++n)
            {
                const int n_id = dirichlet_nodes[n];
                bool found = false;
                for (const auto &bs : bases)
                {
                    for (const auto &b : bs.bases)
                    {
                        for (const auto &lg : b.global())
                        {
                            if (lg.index == n_id)
                            {
                                dirichlet_nodes_position[n] = lg.node;
                                found = true;
                                break;
                            }
                        }
 
                        if (found)
                            break;
                    }
 
                    if (found)
                        break;
                }
 
                assert(found);
            }
 
            neumann_nodes_position.resize(neumann_nodes.size());
            for (int n = 0; n < neumann_nodes.size(); ++n)
            {
                const int n_id = neumann_nodes[n];
                bool found = false;
                for (const auto &bs : bases)
                {
                    for (const auto &b : bs.bases)
                    {
                        for (const auto &lg : b.global())
                        {
                            if (lg.index == n_id)
                            {
                                neumann_nodes_position[n] = lg.node;
                                found = true;
                                break;
                            }
                        }
 
                        if (found)
                            break;
                    }
 
                    if (found)
                        break;
                }
 
                assert(found);
            }
        }
 
        const bool has_neumann = local_neumann_boundary.size() > 0 || local_boundary.size() < prev_b_size;
        use_avg_pressure = !has_neumann;
 
        for (int i = prev_bases; i < n_bases; ++i)
        {
            for (int d = 0; d < problem_dim; ++d)
                boundary_nodes.push_back(i * problem_dim + d);
        }
 
        std::sort(boundary_nodes.begin(), boundary_nodes.end());
        auto it = std::unique(boundary_nodes.begin(), boundary_nodes.end());
        boundary_nodes.resize(std::distance(boundary_nodes.begin(), it));
 
        // for elastic pure periodic problem, find an internal node and force zero dirichlet
        if ((!problem->is_time_dependent() || args["time"]["quasistatic"]) && boundary_nodes.size() == 0 && has_periodic_bc())
        {
            // find an internal node to force zero dirichlet
            std::vector<bool> isboundary(n_bases, false);
            for (const auto &lb : total_local_boundary)
            {
                const int e = lb.element_id();
                for (int i = 0; i < lb.size(); ++i)
                {
                    const auto nodes = bases[e].local_nodes_for_primitive(lb.global_primitive_id(i), *mesh);
 
                    for (int n : nodes)
                        isboundary[bases[e].bases[n].global()[0].index] = true;
                }
            }
            int i = 0;
            for (; i < n_bases; i++)
                if (!isboundary[i]) // (!periodic_bc->is_periodic_dof(i))
                    break;
            if (i >= n_bases)
                log_and_throw_error("Failed to find a non-periodic node!");
            const int actual_dim = problem->is_scalar() ? 1 : mesh->dimension();
            for (int d = 0; d < actual_dim; d++)
            {
                boundary_nodes.push_back(i * actual_dim + d);
            }
            logger().info("Fix solution at node {} to remove singularity due to periodic BC", i);
        }
 
        const auto &curret_bases = geom_bases();
        const int n_samples = 10;
        stats.compute_mesh_size(*mesh, curret_bases, n_samples, args["output"]["advanced"]["curved_mesh_size"]);
        if (starting_min_edge_length < 0)
        {
            starting_min_edge_length = stats.min_edge_length;
        }
        if (starting_max_edge_length < 0)
        {
            starting_max_edge_length = stats.mesh_size;
        }
 
        if (is_contact_enabled())
        {
            min_boundary_edge_length = std::numeric_limits<double>::max();
            for (const auto &edge : collision_mesh.edges().rowwise())
            {
                const VectorNd v0 = collision_mesh.rest_positions().row(edge(0));
                const VectorNd v1 = collision_mesh.rest_positions().row(edge(1));
                min_boundary_edge_length = std::min(min_boundary_edge_length, (v1 - v0).norm());
            }
 
            double dhat = Units::convert(args["contact"]["dhat"], units.length());
            args["contact"]["epsv"] = Units::convert(args["contact"]["epsv"], units.velocity());
            args["contact"]["dhat"] = dhat;
 
            if (!has_dhat && dhat > min_boundary_edge_length)
            {
                args["contact"]["dhat"] = double(args["contact"]["dhat_percentage"]) * min_boundary_edge_length;
                logger().info("dhat set to {}", double(args["contact"]["dhat"]));
            }
            else
            {
                if (dhat > min_boundary_edge_length)
                    logger().warn("dhat larger than min boundary edge, {} > {}", dhat, min_boundary_edge_length);
            }
        }
 
        logger().info("n_bases {}", n_bases);
 
        timings.building_basis_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.building_basis_time);
 
        logger().info("flipped elements {}", stats.n_flipped);
        logger().info("h: {}", stats.mesh_size);
        logger().info("n bases: {}", n_bases);
        logger().info("n pressure bases: {}", n_pressure_bases);
 
        ass_vals_cache.clear();
        mass_ass_vals_cache.clear();
        if (n_bases <= args["solver"]["advanced"]["cache_size"])
        {
            timer.start();
            logger().info("Building cache...");
            ass_vals_cache.init(mesh->is_volume(), bases, curret_bases);
            mass_ass_vals_cache.init(mesh->is_volume(), bases, curret_bases, true);
            if (mixed_assembler != nullptr)
                pressure_ass_vals_cache.init(mesh->is_volume(), pressure_bases, curret_bases);
 
            logger().info(" took {}s", timer.getElapsedTime());
        }
 
        out_geom.build_grid(*mesh, args["output"]["advanced"]["sol_on_grid"]);
 
        if ((!problem->is_time_dependent() || args["time"]["quasistatic"]) && boundary_nodes.empty())
        {
            if (has_periodic_bc())
                logger().warn("(Quasi-)Static problem without Dirichlet nodes, will fix solution at one node to find a unique solution!");
            else
                log_and_throw_error("Static problem need to have some Dirichlet nodes!");
        }
    }
 
    void State::build_polygonal_basis()
    {
        if (!mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
 
        rhs.resize(0, 0);
 
        if (poly_edge_to_data.empty() && polys.empty())
        {
            timings.computing_poly_basis_time = 0;
            return;
        }
 
        igl::Timer timer;
        timer.start();
        logger().info("Computing polygonal basis...");
 
        // std::sort(boundary_nodes.begin(), boundary_nodes.end());
 
        // mixed not supports polygonal bases
        assert(n_pressure_bases == 0 || poly_edge_to_data.size() == 0);
 
        int new_bases = 0;
 
        if (iso_parametric())
        {
            if (mesh->is_volume())
            {
                if (args["space"]["poly_basis_type"] == "MeanValue" || args["space"]["poly_basis_type"] == "Wachspress")
                    logger().error("Barycentric bases not supported in 3D");
                assert(assembler->is_linear());
                new_bases = basis::PolygonalBasis3d::build_bases(
                    *dynamic_cast<LinearAssembler *>(assembler.get()),
                    args["space"]["advanced"]["n_harmonic_samples"],
                    *dynamic_cast<Mesh3D *>(mesh.get()),
                    n_bases,
                    args["space"]["advanced"]["quadrature_order"],
                    args["space"]["advanced"]["mass_quadrature_order"],
                    args["space"]["advanced"]["integral_constraints"],
                    bases,
                    bases,
                    poly_edge_to_data,
                    polys_3d);
            }
            else
            {
                if (args["space"]["poly_basis_type"] == "MeanValue")
                {
                    new_bases = basis::MVPolygonalBasis2d::build_bases(
                        assembler->name(),
                        assembler->is_tensor() ? 2 : 1,
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases,
                        args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        bases, local_boundary, polys);
                }
                else if (args["space"]["poly_basis_type"] == "Wachspress")
                {
                    new_bases = basis::WSPolygonalBasis2d::build_bases(
                        assembler->name(),
                        assembler->is_tensor() ? 2 : 1,
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases,
                        args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        bases, local_boundary, polys);
                }
                else
                {
                    assert(assembler->is_linear());
                    new_bases = basis::PolygonalBasis2d::build_bases(
                        *dynamic_cast<LinearAssembler *>(assembler.get()),
                        args["space"]["advanced"]["n_harmonic_samples"],
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases,
                        args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        args["space"]["advanced"]["integral_constraints"],
                        bases,
                        bases,
                        poly_edge_to_data,
                        polys);
                }
            }
        }
        else
        {
            if (mesh->is_volume())
            {
                if (args["space"]["poly_basis_type"] == "MeanValue" || args["space"]["poly_basis_type"] == "Wachspress")
                {
                    logger().error("Barycentric bases not supported in 3D");
                    throw "not implemented";
                }
                else
                {
                    assert(assembler->is_linear());
                    new_bases = basis::PolygonalBasis3d::build_bases(
                        *dynamic_cast<LinearAssembler *>(assembler.get()),
                        args["space"]["advanced"]["n_harmonic_samples"],
                        *dynamic_cast<Mesh3D *>(mesh.get()),
                        n_bases,
                        args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        args["space"]["advanced"]["integral_constraints"],
                        bases,
                        geom_bases_,
                        poly_edge_to_data,
                        polys_3d);
                }
            }
            else
            {
                if (args["space"]["poly_basis_type"] == "MeanValue")
                {
                    new_bases = basis::MVPolygonalBasis2d::build_bases(
                        assembler->name(),
                        assembler->is_tensor() ? 2 : 1,
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases, args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        bases, local_boundary, polys);
                }
                else if (args["space"]["poly_basis_type"] == "Wachspress")
                {
                    new_bases = basis::WSPolygonalBasis2d::build_bases(
                        assembler->name(),
                        assembler->is_tensor() ? 2 : 1,
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases, args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        bases, local_boundary, polys);
                }
                else
                {
                    assert(assembler->is_linear());
                    new_bases = basis::PolygonalBasis2d::build_bases(
                        *dynamic_cast<LinearAssembler *>(assembler.get()),
                        args["space"]["advanced"]["n_harmonic_samples"],
                        *dynamic_cast<Mesh2D *>(mesh.get()),
                        n_bases,
                        args["space"]["advanced"]["quadrature_order"],
                        args["space"]["advanced"]["mass_quadrature_order"],
                        args["space"]["advanced"]["integral_constraints"],
                        bases,
                        geom_bases_,
                        poly_edge_to_data,
                        polys);
                }
            }
        }
 
        timer.stop();
        timings.computing_poly_basis_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.computing_poly_basis_time);
 
        n_bases += new_bases;
    }
 
    void State::build_periodic_collision_mesh()
    {
        assert(!mesh->is_volume());
        const int dim = mesh->dimension();
        const int n_tiles = 2;
 
        if (mesh->dimension() != 2)
            log_and_throw_error("Periodic collision mesh is only implemented in 2D!");
 
        Eigen::MatrixXd V(n_bases, dim);
        for (const auto &bs : bases)
            for (const auto &b : bs.bases)
                for (const auto &g : b.global())
                    V.row(g.index) = g.node;
 
        Eigen::MatrixXi E = collision_mesh.edges();
        for (int i = 0; i < E.size(); i++)
            E(i) = collision_mesh.to_full_vertex_id(E(i));
 
        Eigen::MatrixXd bbox(V.cols(), 2);
        bbox.col(0) = V.colwise().minCoeff();
        bbox.col(1) = V.colwise().maxCoeff();
 
        // remove boundary edges on periodic BC, buggy
        {
            std::vector<int> ind;
            for (int i = 0; i < E.rows(); i++)
            {
                if (!periodic_bc->is_periodic_dof(E(i, 0)) || !periodic_bc->is_periodic_dof(E(i, 1)))
                    ind.push_back(i);
            }
            E = E(ind, Eigen::all).eval();
        }
 
        Eigen::MatrixXd Vtmp, Vnew;
        Eigen::MatrixXi Etmp, Enew;
        Vtmp.setZero(V.rows() * n_tiles * n_tiles, V.cols());
        Etmp.setZero(E.rows() * n_tiles * n_tiles, E.cols());
 
        Eigen::MatrixXd tile_offset = periodic_bc->get_affine_matrix();
 
        for (int i = 0, idx = 0; i < n_tiles; i++)
        {
            for (int j = 0; j < n_tiles; j++)
            {
                Eigen::Vector2d block_id;
                block_id << i, j;
 
                Vtmp.middleRows(idx * V.rows(), V.rows()) = V;
                // Vtmp.block(idx * V.rows(), 0, V.rows(), 1).array() += tile_offset(0) * i;
                // Vtmp.block(idx * V.rows(), 1, V.rows(), 1).array() += tile_offset(1) * j;
                for (int vid = 0; vid < V.rows(); vid++)
                    Vtmp.block(idx * V.rows() + vid, 0, 1, 2) += (tile_offset * block_id).transpose();
 
                Etmp.middleRows(idx * E.rows(), E.rows()) = E.array() + idx * V.rows();
                idx += 1;
            }
        }
 
        // clean duplicated vertices
        Eigen::VectorXi indices;
        {
            std::vector<int> tmp;
            for (int i = 0; i < V.rows(); i++)
            {
                if (periodic_bc->is_periodic_dof(i))
                    tmp.push_back(i);
            }
 
            indices.resize(tmp.size() * n_tiles * n_tiles);
            for (int i = 0; i < n_tiles * n_tiles; i++)
            {
                indices.segment(i * tmp.size(), tmp.size()) = Eigen::Map<Eigen::VectorXi, Eigen::Unaligned>(tmp.data(), tmp.size());
                indices.segment(i * tmp.size(), tmp.size()).array() += i * V.rows();
            }
        }
 
        Eigen::VectorXi potentially_duplicate_mask(Vtmp.rows());
        potentially_duplicate_mask.setZero();
        potentially_duplicate_mask(indices).array() = 1;
        Eigen::MatrixXd candidates = Vtmp(indices, Eigen::all);
 
        Eigen::VectorXi SVI;
        std::vector<int> SVJ;
        SVI.setConstant(Vtmp.rows(), -1);
        int id = 0;
        const double eps = (bbox.col(1) - bbox.col(0)).maxCoeff() * args["boundary_conditions"]["periodic_boundary"]["tolerance"].get<double>();
        for (int i = 0; i < Vtmp.rows(); i++)
        {
            if (SVI[i] < 0)
            {
                SVI[i] = id;
                SVJ.push_back(i);
                if (potentially_duplicate_mask(i))
                {
                    Eigen::VectorXd diffs = (candidates.rowwise() - Vtmp.row(i)).rowwise().norm();
                    for (int j = 0; j < diffs.size(); j++)
                        if (diffs(j) < eps)
                            SVI[indices[j]] = id;
                }
                id++;
            }
        }
        Vnew = Vtmp(SVJ, Eigen::all);
 
        Enew.resizeLike(Etmp);
        for (int d = 0; d < Etmp.cols(); d++)
            Enew.col(d) = SVI(Etmp.col(d));
 
        std::vector<bool> is_on_surface = ipc::CollisionMesh::construct_is_on_surface(Vnew.rows(), Enew);
 
        Eigen::MatrixXi boundary_triangles;
        Eigen::SparseMatrix<double> displacement_map;
        periodic_collision_mesh = ipc::CollisionMesh(is_on_surface,
                                                     Vnew,
                                                     Enew,
                                                     boundary_triangles,
                                                     displacement_map);
 
        periodic_collision_mesh.init_area_jacobians();
 
        periodic_collision_mesh_to_basis.setConstant(Vnew.rows(), -1);
        for (int i = 0; i < V.rows(); i++)
            for (int j = 0; j < n_tiles * n_tiles; j++)
                periodic_collision_mesh_to_basis(SVI[j * V.rows() + i]) = i;
 
        if (periodic_collision_mesh_to_basis.maxCoeff() + 1 != V.rows())
            log_and_throw_error("Failed to tile mesh!");
    }
 
    void State::build_collision_mesh()
    {
        build_collision_mesh(
            *mesh, n_bases, bases, geom_bases(), total_local_boundary, obstacle,
            args, [this](const std::string &p) { return resolve_input_path(p); },
            in_node_to_node, collision_mesh);
    }
 
    void State::build_collision_mesh(
        const mesh::Mesh &mesh,
        const int n_bases,
        const std::vector<basis::ElementBases> &bases,
        const std::vector<basis::ElementBases> &geom_bases,
        const std::vector<mesh::LocalBoundary> &total_local_boundary,
        const mesh::Obstacle &obstacle,
        const json &args,
        const std::function<std::string(const std::string &)> &resolve_input_path,
        const Eigen::VectorXi &in_node_to_node,
        ipc::CollisionMesh &collision_mesh)
    {
        Eigen::MatrixXd collision_vertices;
        Eigen::VectorXi collision_codim_vids;
        Eigen::MatrixXi collision_edges, collision_triangles;
        std::vector<Eigen::Triplet<double>> displacement_map_entries;
 
        if (args.contains("/contact/collision_mesh"_json_pointer)
            && args.at("/contact/collision_mesh/enabled"_json_pointer).get<bool>())
        {
            const json collision_mesh_args = args.at("/contact/collision_mesh"_json_pointer);
            if (collision_mesh_args.contains("linear_map"))
            {
                assert(displacement_map_entries.empty());
                assert(collision_mesh_args.contains("mesh"));
                const std::string root_path = utils::json_value<std::string>(args, "root_path", "");
                // TODO: handle transformation per geometry
                const json transformation = json_as_array(args["geometry"])[0]["transformation"];
                mesh::load_collision_proxy(
                    utils::resolve_path(collision_mesh_args["mesh"], root_path),
                    utils::resolve_path(collision_mesh_args["linear_map"], root_path),
                    in_node_to_node, transformation, collision_vertices, collision_codim_vids,
                    collision_edges, collision_triangles, displacement_map_entries);
            }
            else
            {
                assert(collision_mesh_args.contains("max_edge_length"));
                logger().debug(
                    "Building collision proxy with max edge length={} ...",
                    collision_mesh_args["max_edge_length"].get<double>());
                igl::Timer timer;
                timer.start();
                build_collision_proxy(
                    bases, geom_bases, total_local_boundary, n_bases, mesh.dimension(),
                    collision_mesh_args["max_edge_length"], collision_vertices,
                    collision_triangles, displacement_map_entries,
                    collision_mesh_args["tessellation_type"]);
                if (collision_triangles.size())
                    igl::edges(collision_triangles, collision_edges);
                timer.stop();
                logger().debug(fmt::format(
                    std::locale("en_US.UTF-8"),
                    "Done (took {:g}s, {:L} vertices, {:L} triangles)",
                    timer.getElapsedTime(),
                    collision_vertices.rows(), collision_triangles.rows()));
            }
        }
        else
        {
            io::OutGeometryData::extract_boundary_mesh(
                mesh, n_bases - obstacle.n_vertices(), bases, total_local_boundary,
                collision_vertices, collision_edges, collision_triangles, displacement_map_entries);
        }
 
        // n_bases already contains the obstacle vertices
        const int num_fe_nodes = n_bases - obstacle.n_vertices();
        const int num_fe_collision_vertices = collision_vertices.rows();
        assert(collision_edges.size() == 0 || collision_edges.maxCoeff() < num_fe_collision_vertices);
        assert(collision_triangles.size() == 0 || collision_triangles.maxCoeff() < num_fe_collision_vertices);
 
        // Append the obstacles to the collision mesh
        if (obstacle.n_vertices() > 0)
        {
            append_rows(collision_vertices, obstacle.v());
            append_rows(collision_codim_vids, obstacle.codim_v().array() + num_fe_collision_vertices);
            append_rows(collision_edges, obstacle.e().array() + num_fe_collision_vertices);
            append_rows(collision_triangles, obstacle.f().array() + num_fe_collision_vertices);
 
            if (!displacement_map_entries.empty())
            {
                displacement_map_entries.reserve(displacement_map_entries.size() + obstacle.n_vertices());
                for (int i = 0; i < obstacle.n_vertices(); i++)
                {
                    displacement_map_entries.emplace_back(num_fe_collision_vertices + i, num_fe_nodes + i, 1.0);
                }
            }
        }
 
        std::vector<bool> is_on_surface = ipc::CollisionMesh::construct_is_on_surface(
            collision_vertices.rows(), collision_edges);
        for (const int vid : collision_codim_vids)
        {
            is_on_surface[vid] = true;
        }
 
        Eigen::SparseMatrix<double> displacement_map;
        if (!displacement_map_entries.empty())
        {
            displacement_map.resize(collision_vertices.rows(), n_bases);
            displacement_map.setFromTriplets(displacement_map_entries.begin(), displacement_map_entries.end());
        }
 
        collision_mesh = ipc::CollisionMesh(
            is_on_surface, collision_vertices, collision_edges, collision_triangles,
            displacement_map);
 
        collision_mesh.can_collide = [&collision_mesh, num_fe_collision_vertices](size_t vi, size_t vj) {
            // obstacles do not collide with other obstacles
            return collision_mesh.to_full_vertex_id(vi) < num_fe_collision_vertices
                   || collision_mesh.to_full_vertex_id(vj) < num_fe_collision_vertices;
        };
 
        collision_mesh.init_area_jacobians();
    }
 
    void State::assemble_mass_mat()
    {
        if (!mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
        if (n_bases <= 0)
        {
            logger().error("Build the bases first!");
            return;
        }
        if (assembler->name() == "OperatorSplitting")
        {
            timings.assembling_stiffness_mat_time = 0;
            avg_mass = 1;
            return;
        }
 
        if (!problem->is_time_dependent())
        {
            avg_mass = 1;
            timings.assembling_mass_mat_time = 0;
            return;
        }
 
        mass.resize(0, 0);
 
        igl::Timer timer;
        timer.start();
        logger().info("Assembling mass mat...");
 
        if (mixed_assembler != nullptr)
        {
            StiffnessMatrix velocity_mass;
            mass_matrix_assembler->assemble(mesh->is_volume(), n_bases, bases, geom_bases(), mass_ass_vals_cache, 0, velocity_mass, true);
 
            std::vector<Eigen::Triplet<double>> mass_blocks;
            mass_blocks.reserve(velocity_mass.nonZeros());
 
            for (int k = 0; k < velocity_mass.outerSize(); ++k)
            {
                for (StiffnessMatrix::InnerIterator it(velocity_mass, k); it; ++it)
                {
                    mass_blocks.emplace_back(it.row(), it.col(), it.value());
                }
            }
 
            mass.resize(n_bases * assembler->size(), n_bases * assembler->size());
            mass.setFromTriplets(mass_blocks.begin(), mass_blocks.end());
            mass.makeCompressed();
        }
        else
        {
            mass_matrix_assembler->assemble(mesh->is_volume(), n_bases, bases, geom_bases(), mass_ass_vals_cache, 0, mass, true);
        }
 
        assert(mass.size() > 0);
 
        avg_mass = 0;
        for (int k = 0; k < mass.outerSize(); ++k)
        {
 
            for (StiffnessMatrix::InnerIterator it(mass, k); it; ++it)
            {
                assert(it.col() == k);
                avg_mass += it.value();
            }
        }
 
        avg_mass /= mass.rows();
        logger().info("average mass {}", avg_mass);
 
        if (args["solver"]["advanced"]["lump_mass_matrix"])
        {
            mass = lump_matrix(mass);
        }
 
        timer.stop();
        timings.assembling_mass_mat_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assembling_mass_mat_time);
 
        stats.nn_zero = mass.nonZeros();
        stats.num_dofs = mass.rows();
        stats.mat_size = (long long)mass.rows() * (long long)mass.cols();
        logger().info("sparsity: {}/{}", stats.nn_zero, stats.mat_size);
    }
 
    std::shared_ptr<RhsAssembler> State::build_rhs_assembler(
        const int n_bases_,
        const std::vector<basis::ElementBases> &bases_,
        const assembler::AssemblyValsCache &ass_vals_cache_) const
    {
        json rhs_solver_params = args["solver"]["linear"];
        if (!rhs_solver_params.contains("Pardiso"))
            rhs_solver_params["Pardiso"] = {};
        rhs_solver_params["Pardiso"]["mtype"] = -2; // matrix type for Pardiso (2 = SPD)
 
        const int size = problem->is_scalar() ? 1 : mesh->dimension();
 
        return std::make_shared<RhsAssembler>(
            *assembler, *mesh, obstacle,
            dirichlet_nodes, neumann_nodes,
            dirichlet_nodes_position, neumann_nodes_position,
            n_bases_, size, bases_, geom_bases(), ass_vals_cache_, *problem,
            args["space"]["advanced"]["bc_method"],
            rhs_solver_params);
    }
 
    std::shared_ptr<PressureAssembler> State::build_pressure_assembler(
        const int n_bases_,
        const std::vector<basis::ElementBases> &bases_) const
    {
        const int size = problem->is_scalar() ? 1 : mesh->dimension();
 
        return std::make_shared<PressureAssembler>(
            *assembler, *mesh, obstacle,
            local_pressure_boundary,
            local_pressure_cavity,
            boundary_nodes,
            primitive_to_node(), node_to_primitive(),
            n_bases_, size, bases_, geom_bases(), *problem);
    }
 
    void State::assemble_rhs()
    {
        if (!mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
        if (n_bases <= 0)
        {
            logger().error("Build the bases first!");
            return;
        }
 
        igl::Timer timer;
 
        json p_params = {};
        p_params["formulation"] = assembler->name();
        p_params["root_path"] = root_path();
        {
            RowVectorNd min, max, delta;
            mesh->bounding_box(min, max);
            delta = (max - min) / 2. + min;
            if (mesh->is_volume())
                p_params["bbox_center"] = {delta(0), delta(1), delta(2)};
            else
                p_params["bbox_center"] = {delta(0), delta(1)};
        }
        problem->set_parameters(p_params);
 
        rhs.resize(0, 0);
 
        timer.start();
        logger().info("Assigning rhs...");
 
        solve_data.rhs_assembler = build_rhs_assembler();
        solve_data.rhs_assembler->assemble(mass_matrix_assembler->density(), rhs);
        rhs *= -1;
 
        // if(problem->is_mixed())
        if (mixed_assembler != nullptr)
        {
            const int prev_size = rhs.size();
            const int n_larger = n_pressure_bases + (use_avg_pressure ? (assembler->is_fluid() ? 1 : 0) : 0);
            rhs.conservativeResize(prev_size + n_larger, rhs.cols());
            if (assembler->name() == "OperatorSplitting")
            {
                timings.assigning_rhs_time = 0;
                return;
            }
            // Divergence free rhs
            if (assembler->name() != "Bilaplacian" || local_neumann_boundary.empty())
            {
                rhs.block(prev_size, 0, n_larger, rhs.cols()).setZero();
            }
            else
            {
                Eigen::MatrixXd tmp(n_pressure_bases, 1);
                tmp.setZero();
 
                std::shared_ptr<RhsAssembler> tmp_rhs_assembler = build_rhs_assembler(
                    n_pressure_bases, pressure_bases, pressure_ass_vals_cache);
 
                tmp_rhs_assembler->set_bc(std::vector<LocalBoundary>(), std::vector<int>(), n_boundary_samples(), local_neumann_boundary, tmp);
                rhs.block(prev_size, 0, n_larger, rhs.cols()) = tmp;
            }
        }
 
        timer.stop();
        timings.assigning_rhs_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assigning_rhs_time);
    }
 
    void State::solve_problem(Eigen::MatrixXd &sol, Eigen::MatrixXd &pressure)
    {
        if (!mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
        if (n_bases <= 0)
        {
            logger().error("Build the bases first!");
            return;
        }
 
        // if (rhs.size() <= 0)
        // {
        //  logger().error("Assemble the rhs first!");
        //  return;
        // }
 
        // sol.resize(0, 0);
        // pressure.resize(0, 0);
        stats.spectrum.setZero();
 
        igl::Timer timer;
        timer.start();
        logger().info("Solving {}", assembler->name());
 
        init_solve(sol, pressure);
 
        if (problem->is_time_dependent())
        {
            const double t0 = args["time"]["t0"];
            const int time_steps = args["time"]["time_steps"];
            const double dt = args["time"]["dt"];
 
            // Pre log the output path for easier watching
            if (args["output"]["advanced"]["save_time_sequence"])
            {
                logger().info("Time sequence of simulation will be written to: \"{}\"",
                              resolve_output_path(args["output"]["paraview"]["file_name"]));
            }
 
            if (assembler->name() == "NavierStokes")
                solve_transient_navier_stokes(time_steps, t0, dt, sol, pressure);
            else if (assembler->name() == "OperatorSplitting")
                solve_transient_navier_stokes_split(time_steps, dt, sol, pressure);
            else if (is_homogenization())
                solve_homogenization(time_steps, t0, dt, sol);
            else if (is_problem_linear())
                solve_transient_linear(time_steps, t0, dt, sol, pressure);
            else if (!assembler->is_linear() && problem->is_scalar())
                throw std::runtime_error("Nonlinear scalar problems are not supported yet!");
            else
                solve_transient_tensor_nonlinear(time_steps, t0, dt, sol);
        }
        else
        {
            if (assembler->name() == "NavierStokes")
                solve_navier_stokes(sol, pressure);
            else if (is_homogenization())
                solve_homogenization(/* time steps */ 0, /* t0 */ 0, /* dt */ 0, sol);
            else if (is_problem_linear())
            {
                init_linear_solve(sol);
                solve_linear(sol, pressure);
                if (optimization_enabled != solver::CacheLevel::None)
                    cache_transient_adjoint_quantities(0, sol, Eigen::MatrixXd::Zero(mesh->dimension(), mesh->dimension()));
            }
            else if (!assembler->is_linear() && problem->is_scalar())
                throw std::runtime_error("Nonlinear scalar problems are not supported yet!");
            else
            {
                init_nonlinear_tensor_solve(sol);
                solve_tensor_nonlinear(sol);
                if (optimization_enabled != solver::CacheLevel::None)
                    cache_transient_adjoint_quantities(0, sol, Eigen::MatrixXd::Zero(mesh->dimension(), mesh->dimension()));
 
                const std::string state_path = resolve_output_path(args["output"]["data"]["state"]);
                if (!state_path.empty())
                    write_matrix(state_path, "u", sol);
            }
        }
 
        timer.stop();
        timings.solving_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.solving_time);
    }
 
} // namespace polyfem