Validity of Lagrange (Bézier) and rational Bézier quads of degree 2
SUMMARYFinite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. Th...
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| Veröffentlicht in: | International journal for numerical methods in engineering 2014-08, Vol.99 (8), p.611-632 |
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| container_title | International journal for numerical methods in engineering |
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| creator | George, P.L. Borouchaki, H. |
| description | SUMMARYFinite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd. |
| doi_str_mv | 10.1002/nme.4696 |
| format | Article |
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This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd.</description><identifier>ISSN: 0029-5981</identifier><identifier>EISSN: 1097-0207</identifier><identifier>DOI: 10.1002/nme.4696</identifier><identifier>CODEN: IJNMBH</identifier><language>eng</language><publisher>Bognor Regis: Blackwell Publishing Ltd</publisher><subject>9-node and 8-node quads ; Bezier ; Bézier curve ; Bézier quad ; Computer Science ; Curves (geometry) ; Finite element method ; high-order element ; Jacobian ; Jacobians ; Mathematical analysis ; Mathematical models ; Numerical Analysis ; Q2 finite element ; Q2 finite element quads ; Q2 mesh ; Quadrilaterals ; rational Bézier curve ; rational Bézier quad</subject><ispartof>International journal for numerical methods in engineering, 2014-08, Vol.99 (8), p.611-632</ispartof><rights>Copyright © 2014 John Wiley & Sons, Ltd.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c3986-8cf8a1454683fdc2ad3d4e6b1851885deccb0fb2788af58174e2893a9ae315913</citedby><cites>FETCH-LOGICAL-c3986-8cf8a1454683fdc2ad3d4e6b1851885deccb0fb2788af58174e2893a9ae315913</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Fnme.4696$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Fnme.4696$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>230,314,778,782,883,1414,27907,27908,45557,45558</link.rule.ids><backlink>$$Uhttps://utt.hal.science/hal-02286200$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>George, P.L.</creatorcontrib><creatorcontrib>Borouchaki, H.</creatorcontrib><title>Validity of Lagrange (Bézier) and rational Bézier quads of degree 2</title><title>International journal for numerical methods in engineering</title><addtitle>Int. J. Numer. Meth. Engng</addtitle><description>SUMMARYFinite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd.</description><subject>9-node and 8-node quads</subject><subject>Bezier</subject><subject>Bézier curve</subject><subject>Bézier quad</subject><subject>Computer Science</subject><subject>Curves (geometry)</subject><subject>Finite element method</subject><subject>high-order element</subject><subject>Jacobian</subject><subject>Jacobians</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Numerical Analysis</subject><subject>Q2 finite element</subject><subject>Q2 finite element quads</subject><subject>Q2 mesh</subject><subject>Quadrilaterals</subject><subject>rational Bézier curve</subject><subject>rational Bézier quad</subject><issn>0029-5981</issn><issn>1097-0207</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNp10NtK5EAQBuBGVnBWBR8h4I1eRPuQPl26MqtiPFx4AG-amqQytmYS7Z5ZHd_I5_DFTBhRXPCqoOrrovonZIPRHUYp320muJMpq5bIgFGrU8qp_kUG3cim0hq2Qn7HeEcpY5KKARleQe1LP50nbZXkMA7QjDHZ-vP2-uIxbCfQlEmAqW8bqJOPbvI4gzL2D0ocB8SEr5HlCuqI6x91lVz-HV7sH6b52cHR_l6eFsIalZqiMsAymSkjqrLgUIoyQzViRjJjZIlFMaLViGtjoJKG6Qy5sQIsoGDSMrFKthd7b6F2D8FPIMxdC94d7uWu71HOjeKU_uvt1sI-hPZxhnHqJj4WWNfQYDuLjinNMmO0MB3d_I_etbPQ_bhTsjuXUW3518IitDEGrD4vYNT12bsue9dn39F0QZ98jfMfnTs9GX73Pk7x-dNDuHdKCy3d9emB0zfqPD_h1B2Ld2Obkb4</recordid><startdate>20140824</startdate><enddate>20140824</enddate><creator>George, P.L.</creator><creator>Borouchaki, H.</creator><general>Blackwell Publishing Ltd</general><general>Wiley Subscription Services, Inc</general><general>Wiley</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope></search><sort><creationdate>20140824</creationdate><title>Validity of Lagrange (Bézier) and rational Bézier quads of degree 2</title><author>George, P.L. ; Borouchaki, H.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3986-8cf8a1454683fdc2ad3d4e6b1851885deccb0fb2788af58174e2893a9ae315913</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>9-node and 8-node quads</topic><topic>Bezier</topic><topic>Bézier curve</topic><topic>Bézier quad</topic><topic>Computer Science</topic><topic>Curves (geometry)</topic><topic>Finite element method</topic><topic>high-order element</topic><topic>Jacobian</topic><topic>Jacobians</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Numerical Analysis</topic><topic>Q2 finite element</topic><topic>Q2 finite element quads</topic><topic>Q2 mesh</topic><topic>Quadrilaterals</topic><topic>rational Bézier curve</topic><topic>rational Bézier quad</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>George, P.L.</creatorcontrib><creatorcontrib>Borouchaki, H.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>International journal for numerical methods in engineering</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>George, P.L.</au><au>Borouchaki, H.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Validity of Lagrange (Bézier) and rational Bézier quads of degree 2</atitle><jtitle>International journal for numerical methods in engineering</jtitle><addtitle>Int. J. Numer. Meth. Engng</addtitle><date>2014-08-24</date><risdate>2014</risdate><volume>99</volume><issue>8</issue><spage>611</spage><epage>632</epage><pages>611-632</pages><issn>0029-5981</issn><eissn>1097-0207</eissn><coden>IJNMBH</coden><abstract>SUMMARYFinite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd.</abstract><cop>Bognor Regis</cop><pub>Blackwell Publishing Ltd</pub><doi>10.1002/nme.4696</doi><tpages>22</tpages></addata></record> |
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| ispartof | International journal for numerical methods in engineering, 2014-08, Vol.99 (8), p.611-632 |
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| source | Wiley Online Library Journals Frontfile Complete |
| subjects | 9-node and 8-node quads Bezier Bézier curve Bézier quad Computer Science Curves (geometry) Finite element method high-order element Jacobian Jacobians Mathematical analysis Mathematical models Numerical Analysis Q2 finite element Q2 finite element quads Q2 mesh Quadrilaterals rational Bézier curve rational Bézier quad |
| title | Validity of Lagrange (Bézier) and rational Bézier quads of degree 2 |
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