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dolfinx/main/cpp/demos/demo_biharmonic.html

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\forall \ K \in \mathcal{T} \right\}
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\]</div>
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<p>and considering the boundary conditions</p>
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<div class="amsmath math notranslate nohighlight" id="equation-545b8464-4952-40e2-8a8c-62ed0dd802b5">
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<span class="eqno">(1)<a class="headerlink" href="#equation-545b8464-4952-40e2-8a8c-62ed0dd802b5" title="Permalink to this equation"></a></span>\[\begin{align}
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<div class="amsmath math notranslate nohighlight" id="equation-5bcb0bbf-4a2b-4b54-9519-f70e695ad288">
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<span class="eqno">(1)<a class="headerlink" href="#equation-5bcb0bbf-4a2b-4b54-9519-f70e695ad288" title="Permalink to this equation"></a></span>\[\begin{align}
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u &amp;= 0 \quad {\rm on} \ \partial\Omega, \\
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\nabla^{2} u &amp;= 0 \quad {\rm on} \ \partial\Omega,
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\end{align}\]</div>

dolfinx/main/cpp/doxygen/d1/d0b/VTKHDF_8h_source.html

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dolfinx/main/cpp/doxygen/d3/d56/vtk__utils_8h_source.html

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<div class="ttc" id="anamespacedolfinx_1_1fem_html"><div class="ttname"><a href="../../d8/dbf/namespacedolfinx_1_1fem.html">dolfinx::fem</a></div><div class="ttdoc">Finite element method functionality.</div><div class="ttdef"><b>Definition</b> assemble_expression_impl.h:23</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1fem_html_a7fca24b826a16ff97db3431784cc8a6bac00f0c4675b91fb8b918e4079a0b1bac"><div class="ttname"><a href="../../d8/dbf/namespacedolfinx_1_1fem.html#a7fca24b826a16ff97db3431784cc8a6bac00f0c4675b91fb8b918e4079a0b1bac">dolfinx::fem::doftransform::standard</a></div><div class="ttdeci">@ standard</div><div class="ttdoc">Standard.</div><div class="ttdef"><b>Definition</b> FiniteElement.h:27</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1fem_html_afc75d761461ccee9de059fca268d90c6"><div class="ttname"><a href="../../d8/dbf/namespacedolfinx_1_1fem.html#afc75d761461ccee9de059fca268d90c6">dolfinx::fem::FunctionSpace</a></div><div class="ttdeci">FunctionSpace(U mesh, V element, W dofmap) -&gt; FunctionSpace&lt; typename std::remove_cvref&lt; typename U::element_type &gt;::type::geometry_type::value_type &gt;</div><div class="ttdoc">Type deduction.</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a7ad2c7416cecb3a39648b223d545a20b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">dolfinx::io::cells::perm_vtk</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_vtk(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from VTK to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:588</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_aab23f42f403e856a22080310440af205"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">dolfinx::io::cells::transpose</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; transpose(std::span&lt; const std::uint16_t &gt; map)</div><div class="ttdoc">Compute the transpose of a re-ordering map.</div><div class="ttdef"><b>Definition</b> cells.cpp:665</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a7ad2c7416cecb3a39648b223d545a20b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">dolfinx::io::cells::perm_vtk</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_vtk(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from VTK to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:516</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_aab23f42f403e856a22080310440af205"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">dolfinx::io::cells::transpose</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; transpose(std::span&lt; const std::uint16_t &gt; map)</div><div class="ttdoc">Compute the transpose of a re-ordering map.</div><div class="ttdef"><b>Definition</b> cells.cpp:670</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_html"><div class="ttname"><a href="../../d6/d7a/namespacedolfinx_1_1io.html">dolfinx::io</a></div><div class="ttdoc">Support for file IO.</div><div class="ttdef"><b>Definition</b> cells.h:119</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_html_aa6f5bca201b37b723b59be5eb44f0454"><div class="ttname"><a href="../../d6/d7a/namespacedolfinx_1_1io.html#aa6f5bca201b37b723b59be5eb44f0454">dolfinx::io::vtk_mesh_from_space</a></div><div class="ttdeci">std::tuple&lt; std::vector&lt; T &gt;, std::array&lt; std::size_t, 2 &gt;, std::vector&lt; std::int64_t &gt;, std::vector&lt; std::uint8_t &gt;, std::vector&lt; std::int64_t &gt;, std::array&lt; std::size_t, 2 &gt; &gt; vtk_mesh_from_space(const fem::FunctionSpace&lt; T &gt; &amp;V)</div><div class="ttdoc">Given a FunctionSpace, create a topology and geometry based on the dof coordinates.</div><div class="ttdef"><b>Definition</b> vtk_utils.h:188</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_html_ad16336343ac549029a3c218f20216d85"><div class="ttname"><a href="../../d6/d7a/namespacedolfinx_1_1io.html#ad16336343ac549029a3c218f20216d85">dolfinx::io::extract_vtk_connectivity</a></div><div class="ttdeci">std::pair&lt; std::vector&lt; std::int64_t &gt;, std::array&lt; std::size_t, 2 &gt; &gt; extract_vtk_connectivity(md::mdspan&lt; const std::int32_t, md::dextents&lt; std::size_t, 2 &gt; &gt; dofmap_x, mesh::CellType cell_type)</div><div class="ttdoc">Extract the cell topology (connectivity) in VTK ordering for all cells the mesh. The &#39;topology&#39; inclu...</div><div class="ttdef"><b>Definition</b> vtk_utils.cpp:23</div></div>

dolfinx/main/cpp/doxygen/d9/de1/cells_8h_source.html

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<div class="foldopen" id="foldopen00118" data-start="{" data-end="}">
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<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"><a class="line" href="../../d6/d7a/namespacedolfinx_1_1io.html"> 118</a></span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html">dolfinx::io::cells</a></div>
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<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span>{</div>
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<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">perm_vtk</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> type, <span class="keywordtype">int</span> num_nodes);</div>
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<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span></div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#acfaaa0934664ccb3c31fb92c2b5a864c">perm_gmsh</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> type, <span class="keywordtype">int</span> num_nodes);</div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span></div>
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<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">transpose</a>(std::span&lt;const std::uint16_t&gt; map);</div>
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<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span></div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span>std::vector&lt;std::int64_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#ab9635231e8fe37b49c28e51435143649">apply_permutation</a>(std::span&lt;const std::int64_t&gt; <a class="code hl_namespace" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html">cells</a>,</div>
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<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> std::array&lt;std::size_t, 2&gt; shape,</div>
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<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> std::span&lt;const std::uint16_t&gt; p);</div>
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<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span></div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span>std::int8_t <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a62012942c581b792bda9cdf9c9f3746b">get_vtk_cell_type</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> cell, <span class="keywordtype">int</span> dim);</div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> </div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span>} <span class="comment">// namespace dolfinx::io::cells</span></div>
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<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span></div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span><span class="keywordtype">int</span> <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#adcb387641551d54386e2246826b3d982">cell_degree</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> type, <span class="keywordtype">int</span> num_nodes);</div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span></div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">perm_vtk</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> type, <span class="keywordtype">int</span> num_nodes);</div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span></div>
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<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#acfaaa0934664ccb3c31fb92c2b5a864c">perm_gmsh</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> type, <span class="keywordtype">int</span> num_nodes);</div>
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<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span></div>
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<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span>std::vector&lt;std::uint16_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">transpose</a>(std::span&lt;const std::uint16_t&gt; map);</div>
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span></div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span>std::vector&lt;std::int64_t&gt; <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#ab9635231e8fe37b49c28e51435143649">apply_permutation</a>(std::span&lt;const std::int64_t&gt; <a class="code hl_namespace" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html">cells</a>,</div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> std::array&lt;std::size_t, 2&gt; shape,</div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> std::span&lt;const std::uint16_t&gt; p);</div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span></div>
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<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span>std::int8_t <a class="code hl_function" href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a62012942c581b792bda9cdf9c9f3746b">get_vtk_cell_type</a>(<a class="code hl_enumeration" href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">mesh::CellType</a> cell, <span class="keywordtype">int</span> dim);</div>
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<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> </div>
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<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span>} <span class="comment">// namespace dolfinx::io::cells</span></div>
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</div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html">dolfinx::io::cells</a></div><div class="ttdoc">Functions for the re-ordering of input mesh topology to the DOLFINx ordering, and transpose orderings...</div><div class="ttdef"><b>Definition</b> cells.h:119</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a62012942c581b792bda9cdf9c9f3746b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a62012942c581b792bda9cdf9c9f3746b">dolfinx::io::cells::get_vtk_cell_type</a></div><div class="ttdeci">std::int8_t get_vtk_cell_type(mesh::CellType cell, int dim)</div><div class="ttdoc">Get VTK cell identifier.</div><div class="ttdef"><b>Definition</b> cells.cpp:693</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a7ad2c7416cecb3a39648b223d545a20b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">dolfinx::io::cells::perm_vtk</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_vtk(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from VTK to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:588</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_aab23f42f403e856a22080310440af205"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">dolfinx::io::cells::transpose</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; transpose(std::span&lt; const std::uint16_t &gt; map)</div><div class="ttdoc">Compute the transpose of a re-ordering map.</div><div class="ttdef"><b>Definition</b> cells.cpp:665</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_ab9635231e8fe37b49c28e51435143649"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#ab9635231e8fe37b49c28e51435143649">dolfinx::io::cells::apply_permutation</a></div><div class="ttdeci">std::vector&lt; std::int64_t &gt; apply_permutation(std::span&lt; const std::int64_t &gt; cells, std::array&lt; std::size_t, 2 &gt; shape, std::span&lt; const std::uint16_t &gt; p)</div><div class="ttdoc">Permute cell topology by applying a permutation array for each cell.</div><div class="ttdef"><b>Definition</b> cells.cpp:674</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_acfaaa0934664ccb3c31fb92c2b5a864c"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#acfaaa0934664ccb3c31fb92c2b5a864c">dolfinx::io::cells::perm_gmsh</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_gmsh(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from Gmsh to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:626</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a62012942c581b792bda9cdf9c9f3746b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a62012942c581b792bda9cdf9c9f3746b">dolfinx::io::cells::get_vtk_cell_type</a></div><div class="ttdeci">std::int8_t get_vtk_cell_type(mesh::CellType cell, int dim)</div><div class="ttdoc">Get VTK cell identifier.</div><div class="ttdef"><b>Definition</b> cells.cpp:698</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_a7ad2c7416cecb3a39648b223d545a20b"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#a7ad2c7416cecb3a39648b223d545a20b">dolfinx::io::cells::perm_vtk</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_vtk(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from VTK to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:516</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_aab23f42f403e856a22080310440af205"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#aab23f42f403e856a22080310440af205">dolfinx::io::cells::transpose</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; transpose(std::span&lt; const std::uint16_t &gt; map)</div><div class="ttdoc">Compute the transpose of a re-ordering map.</div><div class="ttdef"><b>Definition</b> cells.cpp:670</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_ab9635231e8fe37b49c28e51435143649"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#ab9635231e8fe37b49c28e51435143649">dolfinx::io::cells::apply_permutation</a></div><div class="ttdeci">std::vector&lt; std::int64_t &gt; apply_permutation(std::span&lt; const std::int64_t &gt; cells, std::array&lt; std::size_t, 2 &gt; shape, std::span&lt; const std::uint16_t &gt; p)</div><div class="ttdoc">Permute cell topology by applying a permutation array for each cell.</div><div class="ttdef"><b>Definition</b> cells.cpp:679</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_acfaaa0934664ccb3c31fb92c2b5a864c"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#acfaaa0934664ccb3c31fb92c2b5a864c">dolfinx::io::cells::perm_gmsh</a></div><div class="ttdeci">std::vector&lt; std::uint16_t &gt; perm_gmsh(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Permutation array to map from Gmsh to DOLFINx node ordering.</div><div class="ttdef"><b>Definition</b> cells.cpp:554</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1io_1_1cells_html_adcb387641551d54386e2246826b3d982"><div class="ttname"><a href="../../df/dcf/namespacedolfinx_1_1io_1_1cells.html#adcb387641551d54386e2246826b3d982">dolfinx::io::cells::cell_degree</a></div><div class="ttdeci">int cell_degree(mesh::CellType type, int num_nodes)</div><div class="ttdoc">Get the Lagrange order of a given cell with a given number of nodes.</div><div class="ttdef"><b>Definition</b> cells.cpp:592</div></div>
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<div class="ttc" id="anamespacedolfinx_1_1mesh_html_ac3c2cbbef08f3b7ddc3b06c6bd5a2271"><div class="ttname"><a href="../../dd/d7d/namespacedolfinx_1_1mesh.html#ac3c2cbbef08f3b7ddc3b06c6bd5a2271">dolfinx::mesh::CellType</a></div><div class="ttdeci">CellType</div><div class="ttdoc">Cell type identifier.</div><div class="ttdef"><b>Definition</b> cell_types.h:22</div></div>
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