Isogeometric analysis with geometrically continuous functions on two-patch geometries. (October 2015)

• Record Type: Journal Article
• Authors: Kapl, Mario; Vitrih, Vito; Jüttler, Bert; Birner, Katharina
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 7(2015)
• Page Start: 1518
• Page End: 1538
• Publication Date: 2015-10
• Subjects: Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.04.004
• Languages: English
• ISSNs: 0898-1221

Abstract

<abstract xml:lang="en" abstract-type="author" id="a000005"> <title id="st000005">Abstract</title> <sec> <p id="sp000075">We study the linear space of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5n9kh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si108.gif" display="inline" overflow="scroll" id="d13e3547" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-smooth isogeometric functions defined on a multi-patch domain <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5prm8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si109.gif" display="inline" overflow="scroll" id="d13e3556" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>. We show that the construction of these functions is closely related to the concept of geometric continuity of surfaces, which has originated in geometric design. More precisely, the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5n9kh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si108.gif" display="inline" overflow="scroll" id="d13e3569" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (<inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5sn96" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si111.gif" display="inline" overflow="scroll" id="d13e3578" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-smoothness) of their graph surfaces. This motivates us to call them <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5n9kh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si108.gif" display="inline" overflow="scroll" id="d13e3587" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5rq63" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si113.gif" display="inline" overflow="scroll" id="d13e3597" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2nc5mjbs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si104.gif" display="inline" overflow="scroll" id="d13e3606" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> approximation and for solving Poisson's equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.</p> </sec> </abstract>