Geometric validity (positive jacobian) of high-order Lagrange finite elements, theory and practical guidance. (July 2016)

• Record Type: Journal Article
• Authors: George, P.; Borouchaki, H.; Barral, N.
• Is Part Of: Engineering with computers. Volume 32:Number 3(2016)
• Page Start: 405
• Page End: 424
• Publication Date: 2016-07
• Subjects: High-order finite elements -- Bézier curves -- Bézier patches; Engineering design -- Data processing -- Periodicals; Computer-aided design -- Periodicals; Conception technique -- Informatique -- Périodiques; Conception assistée par ordinateur -- Périodiques; Electronic journals; 620.00285
• DOI: 10.1007/s00366-015-0422-1
• Languages: English
• ISSNs: 0177-0667

Abstract

Abstract Finite elements of degree two or more are needed to solve various P.D.E. problems. This paper discusses a method to validate such meshes for the case of the usual Lagrange elements of various degrees. The first section of this paper comes back to Bézier curve and Bézier patches of arbitrary degree. The way in which a Bézier patch and a finite element are related is recalled. The usual Lagrange finite elements of various degrees are discussed, including simplices (triangle and tetrahedron), quads, prisms (pentahedron), pyramids and hexes together with some low-degree Serendipity elements. A validity condition, the positivity of the jacobian, is exhibited for these elements. Elements of various degrees are envisaged also including some "linear" elements (therefore straight-sided elements of degree 1) because the jacobian (polynomial) of some of them is not totally trivial.