MeshHex8.hpp
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Created: Thu, Oct 31, 18:34

MeshHex8.hpp
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	/**
	Implementation of MeshHex8.h

	\file MeshHex8.hpp
	\copyright Copyright 2017. Tom de Geus. All rights reserved.
	\license This project is released under the GNU Public License (GPLv3).
	*/

	#ifndef GOOSEFEM_MESHHEX8_HPP
	#define GOOSEFEM_MESHHEX8_HPP

	#include "MeshHex8.h"

	namespace GooseFEM {
	namespace Mesh {
	namespace Hex8 {

	inline Regular::Regular(size_t nelx, size_t nely, size_t nelz, double h)
	{
	m_h = h;
	m_nelx = nelx;
	m_nely = nely;
	m_nelz = nelz;
	m_nne = 8;
	m_ndim = 3;

	GOOSEFEM_ASSERT(m_nelx >= 1ul);
	GOOSEFEM_ASSERT(m_nely >= 1ul);
	GOOSEFEM_ASSERT(m_nelz >= 1ul);

	m_nnode = (m_nelx + 1) * (m_nely + 1) * (m_nelz + 1);
	m_nelem = m_nelx * m_nely * m_nelz;
	}

	inline ElementType Regular::getElementType_impl() const
	{
	return ElementType::Hex8;
	}

	inline size_t Regular::nelx_impl() const
	{
	return m_nelx;
	}

	inline size_t Regular::nely_impl() const
	{
	return m_nely;
	}

	inline size_t Regular::nelz_impl() const
	{
	return m_nelz;
	}

	inline xt::xtensor<double, 2> Regular::coor_impl() const
	{
	xt::xtensor<double, 2> ret = xt::empty<double>({m_nnode, m_ndim});

	xt::xtensor<double, 1> x =
	xt::linspace<double>(0.0, m_h * static_cast<double>(m_nelx), m_nelx + 1);
	xt::xtensor<double, 1> y =
	xt::linspace<double>(0.0, m_h * static_cast<double>(m_nely), m_nely + 1);
	xt::xtensor<double, 1> z =
	xt::linspace<double>(0.0, m_h * static_cast<double>(m_nelz), m_nelz + 1);

	size_t inode = 0;

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(inode, 0) = x(ix);
	ret(inode, 1) = y(iy);
	ret(inode, 2) = z(iz);
	++inode;
	}
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 2> Regular::conn_impl() const
	{
	xt::xtensor<size_t, 2> ret = xt::empty<size_t>({m_nelem, m_nne});

	size_t ielem = 0;

	for (size_t iz = 0; iz < m_nelz; ++iz) {
	for (size_t iy = 0; iy < m_nely; ++iy) {
	for (size_t ix = 0; ix < m_nelx; ++ix) {
	ret(ielem, 0) = iy * (m_nelx + 1) + ix + iz * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 1) = iy * (m_nelx + 1) + (ix + 1) + iz * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 3) = (iy + 1) * (m_nelx + 1) + ix + iz * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 2) = (iy + 1) * (m_nelx + 1) + (ix + 1) + iz * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 4) = iy * (m_nelx + 1) + ix + (iz + 1) * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 5) = (iy) * (m_nelx + 1) + (ix + 1) + (iz + 1) * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 7) = (iy + 1) * (m_nelx + 1) + ix + (iz + 1) * (m_nely + 1) * (m_nelx + 1);
	ret(ielem, 6) = (iy + 1) * (m_nelx + 1) + (ix + 1) + (iz + 1) * (m_nely + 1) * (m_nelx + 1);
	++ielem;
	}
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFront_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx + 1) * (m_nely + 1)});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(iy * (m_nelx + 1) + ix) = iy * (m_nelx + 1) + ix;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBack_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx + 1) * (m_nely + 1)});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(iy * (m_nelx + 1) + ix) =
	iy * (m_nelx + 1) + ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesLeft_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nely + 1) * (m_nelz + 1)});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iz * (m_nely + 1) + iy) = iy * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesRight_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nely + 1) * (m_nelz + 1)});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iz * (m_nely + 1) + iy) =
	iy * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottom_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx + 1) * (m_nelz + 1)});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(iz * (m_nelx + 1) + ix) = ix + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTop_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx + 1) * (m_nelz + 1)});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(iz * (m_nelx + 1) + ix) =
	ix + m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx - 1) * (m_nely - 1)});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret((iy - 1) * (m_nelx - 1) + (ix - 1)) = iy * (m_nelx + 1) + ix;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx - 1) * (m_nely - 1)});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret((iy - 1) * (m_nelx - 1) + (ix - 1)) =
	iy * (m_nelx + 1) + ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesLeftFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nely - 1) * (m_nelz - 1)});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret((iz - 1) * (m_nely - 1) + (iy - 1)) =
	iy * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesRightFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nely - 1) * (m_nelz - 1)});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret((iz - 1) * (m_nely - 1) + (iy - 1)) =
	iy * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottomFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx - 1) * (m_nelz - 1)});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret((iz - 1) * (m_nelx - 1) + (ix - 1)) = ix + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTopFace_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_nelx - 1) * (m_nelz - 1)});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret((iz - 1) * (m_nelx - 1) + (ix - 1)) =
	ix + m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontBottomEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx + 1});

	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(ix) = ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontTopEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx + 1});

	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(ix) = ix + m_nely * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontLeftEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely + 1});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iy) = iy * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontRightEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely + 1});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iy) = iy * (m_nelx + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackBottomEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx + 1});

	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(ix) = ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackTopEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx + 1});

	for (size_t ix = 0; ix < m_nelx + 1; ++ix) {
	ret(ix) = m_nely * (m_nelx + 1) + ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackLeftEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely + 1});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iy) = iy * (m_nelx + 1) + m_nelz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackRightEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely + 1});

	for (size_t iy = 0; iy < m_nely + 1; ++iy) {
	ret(iy) = iy * (m_nelx + 1) + m_nelz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottomLeftEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz + 1});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	ret(iz) = iz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottomRightEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz + 1});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	ret(iz) = iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTopLeftEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz + 1});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	ret(iz) = m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTopRightEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz + 1});

	for (size_t iz = 0; iz < m_nelz + 1; ++iz) {
	ret(iz) = m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontBottomOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx - 1});

	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret(ix - 1) = ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontTopOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx - 1});

	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret(ix - 1) = ix + m_nely * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontLeftOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely - 1});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret(iy - 1) = iy * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesFrontRightOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely - 1});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret(iy - 1) = iy * (m_nelx + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackBottomOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx - 1});

	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret(ix - 1) = ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackTopOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelx - 1});

	for (size_t ix = 1; ix < m_nelx; ++ix) {
	ret(ix - 1) = m_nely * (m_nelx + 1) + ix + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackLeftOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely - 1});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret(iy - 1) = iy * (m_nelx + 1) + m_nelz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBackRightOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nely - 1});

	for (size_t iy = 1; iy < m_nely; ++iy) {
	ret(iy - 1) = iy * (m_nelx + 1) + m_nelz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottomLeftOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz - 1});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	ret(iz - 1) = iz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesBottomRightOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz - 1});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	ret(iz - 1) = iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTopLeftOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz - 1});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	ret(iz - 1) = m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> Regular::nodesTopRightOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nelz - 1});

	for (size_t iz = 1; iz < m_nelz; ++iz) {
	ret(iz - 1) = m_nely * (m_nelx + 1) + iz * (m_nelx + 1) * (m_nely + 1) + m_nelx;
	}

	return ret;
	}

	inline size_t Regular::nodesFrontBottomLeftCorner_impl() const
	{
	return 0;
	}

	inline size_t Regular::nodesFrontBottomRightCorner_impl() const
	{
	return m_nelx;
	}

	inline size_t Regular::nodesFrontTopLeftCorner_impl() const
	{
	return m_nely * (m_nelx + 1);
	}

	inline size_t Regular::nodesFrontTopRightCorner_impl() const
	{
	return m_nely * (m_nelx + 1) + m_nelx;
	}

	inline size_t Regular::nodesBackBottomLeftCorner_impl() const
	{
	return m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	inline size_t Regular::nodesBackBottomRightCorner_impl() const
	{
	return m_nelx + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	inline size_t Regular::nodesBackTopLeftCorner_impl() const
	{
	return m_nely * (m_nelx + 1) + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	inline size_t Regular::nodesBackTopRightCorner_impl() const
	{
	return m_nely * (m_nelx + 1) + m_nelx + m_nelz * (m_nely + 1) * (m_nelx + 1);
	}

	inline FineLayer::FineLayer(size_t nelx, size_t nely, size_t nelz, double h, size_t nfine)
	{
	m_h = h;
	m_nne = 8;
	m_ndim = 3;

	// basic assumptions
	GOOSEFEM_ASSERT(nelx >= 1ul);
	GOOSEFEM_ASSERT(nely >= 1ul);
	GOOSEFEM_ASSERT(nelz >= 1ul);

	// store basic info
	m_Lx = m_h * static_cast<double>(nelx);
	m_Lz = m_h * static_cast<double>(nelz);

	// compute element size in y-direction (use symmetry, compute upper half)

	// temporary variables
	size_t nmin, ntot;
	xt::xtensor<size_t, 1> nhx = xt::ones<size_t>({nely});
	xt::xtensor<size_t, 1> nhy = xt::ones<size_t>({nely});
	xt::xtensor<size_t, 1> nhz = xt::ones<size_t>({nely});
	xt::xtensor<int, 1> refine = -1 * xt::ones<int>({nely});

	// minimum height in y-direction (half of the height because of symmetry)
	if (nely % 2 == 0) {
	nmin = nely / 2;
	}
	else {
	nmin = (nely + 1) / 2;
	}

	// minimum number of fine layers in y-direction (minimum 1, middle layer part of this half)
	if (nfine % 2 == 0) {
	nfine = nfine / 2 + 1;
	}
	else {
	nfine = (nfine + 1) / 2;
	}

	if (nfine < 1) {
	nfine = 1;
	}

	if (nfine > nmin) {
	nfine = nmin;
	}

	// loop over element layers in y-direction, try to coarsen using these rules:
	// (1) element size in y-direction <= distance to origin in y-direction
	// (2) element size in x-(z-)direction should fit the total number of elements in
	// x-(z-)direction (3) a certain number of layers have the minimum size "1" (are fine)
	for (size_t iy = nfine;;) {
	// initialize current size in y-direction
	if (iy == nfine) {
	ntot = nfine;
	}
	// check to stop
	if (iy >= nely \|\| ntot >= nmin) {
	nely = iy;
	break;
	}

	// rules (1,2) satisfied: coarsen in x-direction (and z-direction)
	if (3 * nhy(iy) <= ntot && nelx % (3 * nhx(iy)) == 0 && ntot + nhy(iy) < nmin) {
	// - process refinement in x-direction
	refine(iy) = 0;
	nhy(iy) *= 2;
	auto vnhy = xt::view(nhy, xt::range(iy + 1, _));
	auto vnhx = xt::view(nhx, xt::range(iy, _));
	vnhy *= 3;
	vnhx *= 3;

	// - rule (2) satisfied: coarsen next element layer in z-direction
	if (iy + 1 < nely && ntot + 2 * nhy(iy) < nmin) {
	if (nelz % (3 * nhz(iy + 1)) == 0) {
	// - update the number of elements in y-direction
	ntot += nhy(iy);
	// - proceed to next element layer in y-direction
	++iy;
	// - process refinement in z-direction
	refine(iy) = 2;
	nhy(iy) = nhy(iy - 1);
	auto vnhz = xt::view(nhz, xt::range(iy, _));
	vnhz *= 3;
	}
	}
	}

	// rules (1,2) satisfied: coarse in z-direction
	else if (3 * nhy(iy) <= ntot && nelz % (3 * nhz(iy)) == 0 && ntot + nhy(iy) < nmin) {
	// - process refinement in z-direction
	refine(iy) = 2;
	nhy(iy) *= 2;
	auto vnhy = xt::view(nhy, xt::range(iy + 1, _));
	auto vnhz = xt::view(nhz, xt::range(iy, _));
	vnhy *= 3;
	vnhz *= 3;
	}

	// update the number of elements in y-direction
	ntot += nhy(iy);
	// proceed to next element layer in y-direction
	++iy;
	// check to stop
	if (iy >= nely \|\| ntot >= nmin) {
	nely = iy;
	break;
	}
	}

	// symmetrize, compute full information

	// allocate mesh constructor parameters
	m_nhx = xt::empty<size_t>({nely * 2 - 1});
	m_nhy = xt::empty<size_t>({nely * 2 - 1});
	m_nhz = xt::empty<size_t>({nely * 2 - 1});
	m_refine = xt::empty<int>({nely * 2 - 1});
	m_layer_nelx = xt::empty<size_t>({nely * 2 - 1});
	m_layer_nelz = xt::empty<size_t>({nely * 2 - 1});
	m_nnd = xt::empty<size_t>({nely * 2});
	m_startElem = xt::empty<size_t>({nely * 2 - 1});
	m_startNode = xt::empty<size_t>({nely * 2});

	// fill
	// - lower half
	for (size_t iy = 0; iy < nely; ++iy) {
	m_nhx(iy) = nhx(nely - iy - 1);
	m_nhy(iy) = nhy(nely - iy - 1);
	m_nhz(iy) = nhz(nely - iy - 1);
	m_refine(iy) = refine(nely - iy - 1);
	}
	// - upper half
	for (size_t iy = 0; iy < nely - 1; ++iy) {
	m_nhx(iy + nely) = nhx(iy + 1);
	m_nhy(iy + nely) = nhy(iy + 1);
	m_nhz(iy + nely) = nhz(iy + 1);
	m_refine(iy + nely) = refine(iy + 1);
	}

	// update size
	nely = m_nhx.size();

	// compute the number of elements per element layer in y-direction
	for (size_t iy = 0; iy < nely; ++iy) {
	m_layer_nelx(iy) = nelx / m_nhx(iy);
	m_layer_nelz(iy) = nelz / m_nhz(iy);
	}

	// compute the number of nodes per node layer in y-direction
	// - bottom half
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy)
	m_nnd(iy) = (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy) + 1);
	// - top half
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy)
	m_nnd(iy + 1) = (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy) + 1);

	// compute mesh dimensions

	// initialize
	m_nnode = 0;
	m_nelem = 0;
	m_startNode(0) = 0;

	// loop over element layers (bottom -> middle, elements become finer)
	for (size_t i = 0; i < (nely - 1) / 2; ++i) {
	// - store the first element of the layer
	m_startElem(i) = m_nelem;
	// - add the nodes of this layer
	if (m_refine(i) == 0) {
	m_nnode += (3 * m_layer_nelx(i) + 1) * (m_layer_nelz(i) + 1);
	}
	else if (m_refine(i) == 2) {
	m_nnode += (m_layer_nelx(i) + 1) * (3 * m_layer_nelz(i) + 1);
	}
	else {
	m_nnode += (m_layer_nelx(i) + 1) * (m_layer_nelz(i) + 1);
	}
	// - add the elements of this layer
	if (m_refine(i) == 0) {
	m_nelem += (4 * m_layer_nelx(i)) * (m_layer_nelz(i));
	}
	else if (m_refine(i) == 2) {
	m_nelem += (m_layer_nelx(i)) * (4 * m_layer_nelz(i));
	}
	else {
	m_nelem += (m_layer_nelx(i)) * (m_layer_nelz(i));
	}
	// - store the starting node of the next layer
	m_startNode(i + 1) = m_nnode;
	}

	// loop over element layers (middle -> top, elements become coarser)
	for (size_t i = (nely - 1) / 2; i < nely; ++i) {
	// - store the first element of the layer
	m_startElem(i) = m_nelem;
	// - add the nodes of this layer
	if (m_refine(i) == 0) {
	m_nnode += (5 * m_layer_nelx(i) + 1) * (m_layer_nelz(i) + 1);
	}
	else if (m_refine(i) == 2) {
	m_nnode += (m_layer_nelx(i) + 1) * (5 * m_layer_nelz(i) + 1);
	}
	else {
	m_nnode += (m_layer_nelx(i) + 1) * (m_layer_nelz(i) + 1);
	}
	// - add the elements of this layer
	if (m_refine(i) == 0) {
	m_nelem += (4 * m_layer_nelx(i)) * (m_layer_nelz(i));
	}
	else if (m_refine(i) == 2) {
	m_nelem += (m_layer_nelx(i)) * (4 * m_layer_nelz(i));
	}
	else {
	m_nelem += (m_layer_nelx(i)) * (m_layer_nelz(i));
	}
	// - store the starting node of the next layer
	m_startNode(i + 1) = m_nnode;
	}
	// - add the top row of nodes
	m_nnode += (m_layer_nelx(nely - 1) + 1) * (m_layer_nelz(nely - 1) + 1);
	}

	inline size_t FineLayer::nelx_impl() const
	{
	return xt::amax(m_layer_nelx)();
	}

	inline size_t FineLayer::nely_impl() const
	{
	return xt::sum(m_nhy)();
	}

	inline size_t FineLayer::nelz_impl() const
	{
	return xt::amax(m_layer_nelz)();
	}

	inline ElementType FineLayer::getElementType_impl() const
	{
	return ElementType::Hex8;
	}

	inline xt::xtensor<double, 2> FineLayer::coor_impl() const
	{
	// allocate output
	xt::xtensor<double, 2> ret = xt::empty<double>({m_nnode, m_ndim});

	// current node, number of element layers
	size_t inode = 0;
	size_t nely = static_cast<size_t>(m_nhy.size());

	// y-position of each main node layer (i.e. excluding node layers for refinement/coarsening)
	// - allocate
	xt::xtensor<double, 1> y = xt::empty<double>({nely + 1});
	// - initialize
	y(0) = 0.0;
	// - compute
	for (size_t iy = 1; iy < nely + 1; ++iy) {
	y(iy) = y(iy - 1) + m_nhy(iy - 1) * m_h;
	}

	// loop over element layers (bottom -> middle) : add bottom layer (+ refinement layer) of nodes

	for (size_t iy = 0;; ++iy) {
	// get positions along the x- and z-axis
	xt::xtensor<double, 1> x = xt::linspace<double>(0.0, m_Lx, m_layer_nelx(iy) + 1);
	xt::xtensor<double, 1> z = xt::linspace<double>(0.0, m_Lz, m_layer_nelz(iy) + 1);

	// add nodes of the bottom layer of this element
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(inode, 0) = x(ix);
	ret(inode, 1) = y(iy);
	ret(inode, 2) = z(iz);
	++inode;
	}
	}

	// stop at middle layer
	if (iy == (nely - 1) / 2) {
	break;
	}

	// add extra nodes of the intermediate layer, for refinement in x-direction
	if (m_refine(iy) == 0) {
	// - get position offset in x- and y-direction
	double dx = m_h * static_cast<double>(m_nhx(iy) / 3);
	double dy = m_h * static_cast<double>(m_nhy(iy) / 2);
	// - add nodes of the intermediate layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	for (size_t j = 0; j < 2; ++j) {
	ret(inode, 0) = x(ix) + dx * static_cast<double>(j + 1);
	ret(inode, 1) = y(iy) + dy;
	ret(inode, 2) = z(iz);
	++inode;
	}
	}
	}
	}

	// add extra nodes of the intermediate layer, for refinement in z-direction
	else if (m_refine(iy) == 2) {
	// - get position offset in y- and z-direction
	double dz = m_h * static_cast<double>(m_nhz(iy) / 3);
	double dy = m_h * static_cast<double>(m_nhy(iy) / 2);
	// - add nodes of the intermediate layer
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t j = 0; j < 2; ++j) {
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(inode, 0) = x(ix);
	ret(inode, 1) = y(iy) + dy;
	ret(inode, 2) = z(iz) + dz * static_cast<double>(j + 1);
	++inode;
	}
	}
	}
	}
	}

	// loop over element layers (middle -> top) : add (refinement layer +) top layer of nodes

	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	// get positions along the x- and z-axis
	xt::xtensor<double, 1> x = xt::linspace<double>(0.0, m_Lx, m_layer_nelx(iy) + 1);
	xt::xtensor<double, 1> z = xt::linspace<double>(0.0, m_Lz, m_layer_nelz(iy) + 1);

	// add extra nodes of the intermediate layer, for refinement in x-direction
	if (m_refine(iy) == 0) {
	// - get position offset in x- and y-direction
	double dx = m_h * static_cast<double>(m_nhx(iy) / 3);
	double dy = m_h * static_cast<double>(m_nhy(iy) / 2);
	// - add nodes of the intermediate layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	for (size_t j = 0; j < 2; ++j) {
	ret(inode, 0) = x(ix) + dx * static_cast<double>(j + 1);
	ret(inode, 1) = y(iy) + dy;
	ret(inode, 2) = z(iz);
	++inode;
	}
	}
	}
	}

	// add extra nodes of the intermediate layer, for refinement in z-direction
	else if (m_refine(iy) == 2) {
	// - get position offset in y- and z-direction
	double dz = m_h * static_cast<double>(m_nhz(iy) / 3);
	double dy = m_h * static_cast<double>(m_nhy(iy) / 2);
	// - add nodes of the intermediate layer
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t j = 0; j < 2; ++j) {
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(inode, 0) = x(ix);
	ret(inode, 1) = y(iy) + dy;
	ret(inode, 2) = z(iz) + dz * static_cast<double>(j + 1);
	++inode;
	}
	}
	}
	}

	// add nodes of the top layer of this element
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(inode, 0) = x(ix);
	ret(inode, 1) = y(iy + 1);
	ret(inode, 2) = z(iz);
	++inode;
	}
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 2> FineLayer::conn_impl() const
	{
	// allocate output
	xt::xtensor<size_t, 2> ret = xt::empty<size_t>({m_nelem, m_nne});

	// current element, number of element layers, starting nodes of each node layer
	size_t ielem = 0;
	size_t nely = static_cast<size_t>(m_nhy.size());
	size_t bot, mid, top;

	// loop over all element layers
	for (size_t iy = 0; iy < nely; ++iy) {
	// - get: starting nodes of bottom(, middle) and top layer
	bot = m_startNode(iy);
	mid = m_startNode(iy) + m_nnd(iy);
	top = m_startNode(iy + 1);

	// - define connectivity: no coarsening/refinement
	if (m_refine(iy) == -1) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	ret(ielem, 0) = bot + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	}
	}
	}

	// - define connectivity: refinement along the x-direction (below the middle layer)
	else if (m_refine(iy) == 0 && iy <= (nely - 1) / 2) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	// -- bottom element
	ret(ielem, 0) = bot + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 3) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 4) = bot + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 7) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ielem++;
	// -- top-right element
	ret(ielem, 0) = bot + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = top + (3 * ix + 3) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (3 * ix + 2) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 3) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 4) = bot + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = top + (3 * ix + 3) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (3 * ix + 2) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 7) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ielem++;
	// -- top-center element
	ret(ielem, 0) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 1) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 2) = top + (3 * ix + 2) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + (3 * ix + 1) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 5) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 6) = top + (3 * ix + 2) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + (3 * ix + 1) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ielem++;
	// -- top-left element
	ret(ielem, 0) = bot + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 2) = top + (3 * ix + 1) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + (3 * ix) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 6) = top + (3 * ix + 1) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + (3 * ix) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ielem++;
	}
	}
	}

	// - define connectivity: coarsening along the x-direction (above the middle layer)
	else if (m_refine(iy) == 0 && iy > (nely - 1) / 2) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	// -- lower-left element
	ret(ielem, 0) = bot + (3 * ix) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (3 * ix + 1) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 2) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 3) = top + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + (3 * ix) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (3 * ix + 1) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 7) = top + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- lower-center element
	ret(ielem, 0) = bot + (3 * ix + 1) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (3 * ix + 2) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 2) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 3) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 4) = bot + (3 * ix + 1) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (3 * ix + 2) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 7) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ielem++;
	// -- lower-right element
	ret(ielem, 0) = bot + (3 * ix + 2) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (3 * ix + 3) + iz * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 4) = bot + (3 * ix + 2) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (3 * ix + 3) + (iz + 1) * (3 * m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ielem++;
	// -- upper element
	ret(ielem, 0) = mid + (2 * ix) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 1) = mid + (2 * ix + 1) + iz * (2 * m_layer_nelx(iy));
	ret(ielem, 2) = top + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + (2 * ix) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 5) = mid + (2 * ix + 1) + (iz + 1) * (2 * m_layer_nelx(iy));
	ret(ielem, 6) = top + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	}
	}
	}

	// - define connectivity: refinement along the z-direction (below the middle layer)
	else if (m_refine(iy) == 2 && iy <= (nely - 1) / 2) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	// -- bottom element
	ret(ielem, 0) = bot + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = bot + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = bot + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + ix + 2 * iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = mid + (ix + 1) + 2 * iz * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- top-back element
	ret(ielem, 0) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (3 * iz + 3) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (3 * iz + 3) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- top-center element
	ret(ielem, 0) = mid + ix + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = mid + (ix + 1) + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- top-front element
	ret(ielem, 0) = bot + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + (3 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + (3 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + ix + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = mid + (ix + 1) + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	}
	}
	}

	// - define connectivity: coarsening along the z-direction (above the middle layer)
	else if (m_refine(iy) == 2 && iy > (nely - 1) / 2) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	// -- bottom-front element
	ret(ielem, 0) = bot + ix + (3 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + (3 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (ix + 1) + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = mid + ix + (2 * iz) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- bottom-center element
	ret(ielem, 0) = bot + ix + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + (3 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = mid + (ix + 1) + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = mid + ix + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- bottom-back element
	ret(ielem, 0) = bot + ix + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = bot + (ix + 1) + (3 * iz + 2) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = bot + ix + (3 * iz + 3) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = bot + (ix + 1) + (3 * iz + 3) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	// -- top element
	ret(ielem, 0) = mid + ix + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 1) = mid + (ix + 1) + (2 * iz) * (m_layer_nelx(iy) + 1);
	ret(ielem, 2) = top + (ix + 1) + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 3) = top + ix + iz * (m_layer_nelx(iy) + 1);
	ret(ielem, 4) = mid + ix + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 5) = mid + (ix + 1) + (2 * iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 6) = top + (ix + 1) + (iz + 1) * (m_layer_nelx(iy) + 1);
	ret(ielem, 7) = top + ix + (iz + 1) * (m_layer_nelx(iy) + 1);
	ielem++;
	}
	}
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::elementsMiddleLayer() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());
	size_t iy = (nely - 1) / 2;

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(iy) * m_layer_nelz(iy)});

	for (size_t ix = 0; ix < m_layer_nelx(iy); ++ix) {
	for (size_t iz = 0; iz < m_layer_nelz(iy); ++iz) {
	ret(ix + iz * m_layer_nelx(iy)) = m_startElem(iy) + ix + iz * m_layer_nelx(iy);
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFront_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelx(iy) + 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelx(iy) + 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(j) = m_startNode(iy) + ix;
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(j) = m_startNode(iy + 1) + ix;
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBack_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelx(iy) + 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelx(iy) + 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(j) = m_startNode(iy) + ix + (m_layer_nelx(iy) + 1) * m_layer_nelz(iy);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy) + 2 * m_layer_nelx(iy) * m_layer_nelz(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy) + 2 * m_layer_nelx(iy) * m_layer_nelz(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t ix = 0; ix < m_layer_nelx(iy) + 1; ++ix) {
	ret(j) = m_startNode(iy + 1) + ix + (m_layer_nelx(iy) + 1) * m_layer_nelz(iy);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesLeft_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelz(iy) + 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 + 1;
	}
	else {
	n += m_layer_nelz(iy) + 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_nnd(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_nnd(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	ret(j) = m_startNode(iy + 1) + iz * (m_layer_nelx(iy) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesRight_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 2)
	n += m_layer_nelz(iy) * 3 + 1;
	else
	n += m_layer_nelz(iy) + 1;
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	if (m_refine(iy) == 2)
	n += m_layer_nelz(iy) * 3 + 1;
	else
	n += m_layer_nelz(iy) + 1;
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + m_nnd(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + m_nnd(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t iz = 0; iz < m_layer_nelz(iy) + 1; ++iz) {
	ret(j) = m_startNode(iy + 1) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottom_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// allocate node list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nnd(nely)});

	// counter
	size_t j = 0;

	// fill node list
	for (size_t ix = 0; ix < m_layer_nelx(0) + 1; ++ix) {
	for (size_t iz = 0; iz < m_layer_nelz(0) + 1; ++iz) {
	ret(j) = m_startNode(0) + ix + iz * (m_layer_nelx(0) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTop_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// allocate node list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_nnd(nely)});

	// counter
	size_t j = 0;

	// fill node list
	for (size_t ix = 0; ix < m_layer_nelx(nely - 1) + 1; ++ix) {
	for (size_t iz = 0; iz < m_layer_nelz(nely - 1) + 1; ++iz) {
	ret(j) = m_startNode(nely) + ix + iz * (m_layer_nelx(nely - 1) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontFace_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelx(iy) - 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelx(iy) - 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t ix = 1; ix < m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix;
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t ix = 1; ix < m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy + 1) + ix;
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackFace_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelx(iy) - 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	if (m_refine(iy) == 0) {
	n += m_layer_nelx(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelx(iy) - 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t ix = 1; ix < m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + (m_layer_nelx(iy) + 1) * m_layer_nelz(iy);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy) + 2 * m_layer_nelx(iy) * m_layer_nelz(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 0) {
	for (size_t ix = 0; ix < 2 * m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy) + ix + m_nnd(iy) + 2 * m_layer_nelx(iy) * m_layer_nelz(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t ix = 1; ix < m_layer_nelx(iy); ++ix) {
	ret(j) = m_startNode(iy + 1) + ix + (m_layer_nelx(iy) + 1) * m_layer_nelz(iy);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesLeftFace_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelz(iy) - 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelz(iy) - 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t iz = 1; iz < m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_nnd(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_nnd(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t iz = 1; iz < m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy + 1) + iz * (m_layer_nelx(iy) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesRightFace_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// number of boundary nodes
	// - initialize
	size_t n = 0;
	// - bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelz(iy) - 1;
	}
	}
	// - top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	if (m_refine(iy) == 2) {
	n += m_layer_nelz(iy) * 3 - 1;
	}
	else {
	n += m_layer_nelz(iy) - 1;
	}
	}

	// allocate node-list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({n});

	// initialize counter: current index in the node-list "ret"
	size_t j = 0;

	// bottom half: bottom node layer (+ middle node layer)
	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	// -- bottom node layer
	for (size_t iz = 1; iz < m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + m_nnd(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}
	}

	// top half: (middle node layer +) top node layer
	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	// -- refinement layer
	if (m_refine(iy) == 2) {
	for (size_t iz = 0; iz < 2 * m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy) + m_nnd(iy) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}
	// -- top node layer
	for (size_t iz = 1; iz < m_layer_nelz(iy); ++iz) {
	ret(j) = m_startNode(iy + 1) + iz * (m_layer_nelx(iy) + 1) + m_layer_nelx(iy);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottomFace_impl() const
	{
	// allocate node list
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({(m_layer_nelx(0) - 1) * (m_layer_nelz(0) - 1)});

	// counter
	size_t j = 0;

	// fill node list
	for (size_t ix = 1; ix < m_layer_nelx(0); ++ix) {
	for (size_t iz = 1; iz < m_layer_nelz(0); ++iz) {
	ret(j) = m_startNode(0) + ix + iz * (m_layer_nelx(0) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTopFace_impl() const
	{
	// number of element layers in y-direction
	size_t nely = static_cast<size_t>(m_nhy.size());

	// allocate node list
	xt::xtensor<size_t, 1> ret =
	xt::empty<size_t>({(m_layer_nelx(nely - 1) - 1) * (m_layer_nelz(nely - 1) - 1)});

	// counter
	size_t j = 0;

	// fill node list
	for (size_t ix = 1; ix < m_layer_nelx(nely - 1); ++ix) {
	for (size_t iz = 1; iz < m_layer_nelz(nely - 1); ++iz) {
	ret(j) = m_startNode(nely) + ix + iz * (m_layer_nelx(nely - 1) + 1);
	++j;
	}
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontBottomEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(0) + 1});

	for (size_t ix = 0; ix < m_layer_nelx(0) + 1; ++ix) {
	ret(ix) = m_startNode(0) + ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontTopEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(nely - 1) + 1});

	for (size_t ix = 0; ix < m_layer_nelx(nely - 1) + 1; ++ix) {
	ret(ix) = m_startNode(nely) + ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontLeftEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely + 1});

	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	ret(iy) = m_startNode(iy);
	}

	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	ret(iy + 1) = m_startNode(iy + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontRightEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely + 1});

	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	ret(iy) = m_startNode(iy) + m_layer_nelx(iy);
	}

	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	ret(iy + 1) = m_startNode(iy + 1) + m_layer_nelx(iy);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackBottomEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(0) + 1});

	for (size_t ix = 0; ix < m_layer_nelx(0) + 1; ++ix) {
	ret(ix) = m_startNode(0) + ix + (m_layer_nelx(0) + 1) * (m_layer_nelz(0));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackTopEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(nely - 1) + 1});

	for (size_t ix = 0; ix < m_layer_nelx(nely - 1) + 1; ++ix) {
	ret(ix) = m_startNode(nely) + ix + (m_layer_nelx(nely - 1) + 1) * (m_layer_nelz(nely - 1));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackLeftEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely + 1});

	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	ret(iy) = m_startNode(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	ret(iy + 1) = m_startNode(iy + 1) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackRightEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely + 1});

	for (size_t iy = 0; iy < (nely + 1) / 2; ++iy) {
	ret(iy) = m_startNode(iy) + m_layer_nelx(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	for (size_t iy = (nely - 1) / 2; iy < nely; ++iy) {
	ret(iy + 1) = m_startNode(iy + 1) + m_layer_nelx(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottomLeftEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(0) + 1});

	for (size_t iz = 0; iz < m_layer_nelz(0) + 1; ++iz) {
	ret(iz) = m_startNode(0) + iz * (m_layer_nelx(0) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottomRightEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(0) + 1});

	for (size_t iz = 0; iz < m_layer_nelz(0) + 1; ++iz) {
	ret(iz) = m_startNode(0) + m_layer_nelx(0) + iz * (m_layer_nelx(0) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTopLeftEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(nely - 1) + 1});

	for (size_t iz = 0; iz < m_layer_nelz(nely - 1) + 1; ++iz) {
	ret(iz) = m_startNode(nely) + iz * (m_layer_nelx(nely - 1) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTopRightEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(nely - 1) + 1});

	for (size_t iz = 0; iz < m_layer_nelz(nely - 1) + 1; ++iz) {
	ret(iz) = m_startNode(nely) + m_layer_nelx(nely - 1) + iz * (m_layer_nelx(nely - 1) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontBottomOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(0) - 1});

	for (size_t ix = 1; ix < m_layer_nelx(0); ++ix) {
	ret(ix - 1) = m_startNode(0) + ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontTopOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(nely - 1) - 1});

	for (size_t ix = 1; ix < m_layer_nelx(nely - 1); ++ix) {
	ret(ix - 1) = m_startNode(nely) + ix;
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontLeftOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely - 1});

	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	ret(iy - 1) = m_startNode(iy);
	}

	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	ret(iy) = m_startNode(iy + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesFrontRightOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely - 1});

	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	ret(iy - 1) = m_startNode(iy) + m_layer_nelx(iy);
	}

	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	ret(iy) = m_startNode(iy + 1) + m_layer_nelx(iy);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackBottomOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(0) - 1});

	for (size_t ix = 1; ix < m_layer_nelx(0); ++ix) {
	ret(ix - 1) = m_startNode(0) + ix + (m_layer_nelx(0) + 1) * (m_layer_nelz(0));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackTopOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelx(nely - 1) - 1});

	for (size_t ix = 1; ix < m_layer_nelx(nely - 1); ++ix) {
	ret(ix - 1) = m_startNode(nely) + ix + (m_layer_nelx(nely - 1) + 1) * (m_layer_nelz(nely - 1));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackLeftOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely - 1});

	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	ret(iy - 1) = m_startNode(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	ret(iy) = m_startNode(iy + 1) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBackRightOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({nely - 1});

	for (size_t iy = 1; iy < (nely + 1) / 2; ++iy) {
	ret(iy - 1) = m_startNode(iy) + m_layer_nelx(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	for (size_t iy = (nely - 1) / 2; iy < nely - 1; ++iy) {
	ret(iy) = m_startNode(iy + 1) + m_layer_nelx(iy) + (m_layer_nelx(iy) + 1) * (m_layer_nelz(iy));
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottomLeftOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(0) - 1});

	for (size_t iz = 1; iz < m_layer_nelz(0); ++iz) {
	ret(iz - 1) = m_startNode(0) + iz * (m_layer_nelx(0) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesBottomRightOpenEdge_impl() const
	{
	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(0) - 1});

	for (size_t iz = 1; iz < m_layer_nelz(0); ++iz) {
	ret(iz - 1) = m_startNode(0) + m_layer_nelx(0) + iz * (m_layer_nelx(0) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTopLeftOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(nely - 1) - 1});

	for (size_t iz = 1; iz < m_layer_nelz(nely - 1); ++iz) {
	ret(iz - 1) = m_startNode(nely) + iz * (m_layer_nelx(nely - 1) + 1);
	}

	return ret;
	}

	inline xt::xtensor<size_t, 1> FineLayer::nodesTopRightOpenEdge_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());

	xt::xtensor<size_t, 1> ret = xt::empty<size_t>({m_layer_nelz(nely - 1) - 1});

	for (size_t iz = 1; iz < m_layer_nelz(nely - 1); ++iz) {
	ret(iz - 1) = m_startNode(nely) + m_layer_nelx(nely - 1) + iz * (m_layer_nelx(nely - 1) + 1);
	}

	return ret;
	}

	inline size_t FineLayer::nodesFrontBottomLeftCorner_impl() const
	{
	return m_startNode(0);
	}

	inline size_t FineLayer::nodesFrontBottomRightCorner_impl() const
	{
	return m_startNode(0) + m_layer_nelx(0);
	}

	inline size_t FineLayer::nodesFrontTopLeftCorner_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());
	return m_startNode(nely);
	}

	inline size_t FineLayer::nodesFrontTopRightCorner_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());
	return m_startNode(nely) + m_layer_nelx(nely - 1);
	}

	inline size_t FineLayer::nodesBackBottomLeftCorner_impl() const
	{
	return m_startNode(0) + (m_layer_nelx(0) + 1) * (m_layer_nelz(0));
	}

	inline size_t FineLayer::nodesBackBottomRightCorner_impl() const
	{
	return m_startNode(0) + m_layer_nelx(0) + (m_layer_nelx(0) + 1) * (m_layer_nelz(0));
	}

	inline size_t FineLayer::nodesBackTopLeftCorner_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());
	return m_startNode(nely) + (m_layer_nelx(nely - 1) + 1) * (m_layer_nelz(nely - 1));
	}

	inline size_t FineLayer::nodesBackTopRightCorner_impl() const
	{
	size_t nely = static_cast<size_t>(m_nhy.size());
	return m_startNode(nely) + m_layer_nelx(nely - 1) + (m_layer_nelx(nely - 1) + 1) * (m_layer_nelz(nely - 1));
	}

	} // namespace Hex8
	} // namespace Mesh
	} // namespace GooseFEM

	#endif

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