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Engineering with Computers
Published in: Engineering with Computers 3/2016

01-07-2016 | Original Article

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

Authors: P. L. George, H. Borouchaki, N. Barral

Published in: Engineering with Computers | Issue 3/2016

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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.
previous article A study into the potential of GPUs for the efficient construction and evaluation of Kriging models
next article Improving the quality of finite volume meshes through genetic optimisation

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Appendix
Available only for authorised users
Footnotes
1
incomplete element can also be written in this way but it is more subtle.
 
2
The true coefficients are \(N_{IJ} = 4 \, Q_{IJ}\).
 
3
note that this is exactly the same form as the element, this fact is true only for the degree 2.
 
4
While not being necessary, we consider the case where the degree is the same in both directions.
 
5
actually, for some reduced elements, one or several internal nodes of the complete element are retained as nodes for the reduced element.
 
6
cf. infra.
 
7
and the way in which they are constructed, a paper being currently under preparation to do this
 
8
the so-called Serendipity relation.
 
9
This number of terms is exactly the number of combinations between the triples of all the vectors that can be constructed with the vertices of the element, e.g. 4 with respect to (u, v, w) and 3 with respect to t, therefore \(4 \times 3\). Note that this property holds for all the elements and whatever the degree.
 
Literature
1.
2.
Babuska I, Guo BQ (1988) The h-p version of the finite element method for domain with curved boundaries. SIAM J Numer Anal 25(4):837–861MathSciNetCrossRefMATH
3.
Babuska I, Guo BQ (1996) Approximation properties of the h-p version of the finite element method. Comput Method Appl Mech Eng 133:319–346MathSciNetCrossRefMATH
4.
Bernadi C, Maday Y, Rapetti F (2004) Discrétisation variationnelles de problèmes aux limites elliptiques, collection mathématiques et applications. Springer, Heidelberg
5.
Bézier P (1986) Courbes et surfaces, Mathématiques et CAO. Hermès, ParisMATH
6.
Borouchaki H, Georg PL (2000) Quality mesh generation, concise review paper. C R Acad Sci Ser IIB 328:505–518
7.
8.
Ciarlet PG (1978) The finite element method. North-Holland, AmsterdamMATH
9.
Ciarlet PG (1991) Basic error estimates for elliptic problems. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol II, finite element methods (Part 1). North-Holland, Amsterdam, pp 17–352CrossRef
10.
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis. Toward integration of CAD and FEA. Wiley, ChichesterCrossRef
11.
Dey S, O’Bara RM, Shephard MS (1999) Curvilinear mesh generation in 3D, In: 8th International meshing roundtable, South Lake Tahoe
12.
Farin G (2002) Curves and surfaces for CAGD. A practical guide, 5th edn. Academic Press, London
13.
Floater MS, Gillette A (2015) Nodal bases for the serendipity family of finite elements. Found Comput Math (to appear)
14.
15.
George PL, Borouchaki H (2014) Sur les éléments finis de Lagrange pyramidaux. RR INRIA 8525
16.
George PL, Borouchaki H (2014) Validity of Lagrange (Bézier) and rational Bézier quads of degree 2. Int J Numer Method Eng 99:611–632MathSciNetCrossRef
17.
George PL, Borouchaki H (2016) Construction and geometric validity (positive jacobian) of serendipity Lagrange finite elements, theory and practical guidance. M2AN (Math Model Numer Anal) (to appear)
18.
Gordon WJ, Hall CA (1973) Construction of curvilinear co-ordinate systems and applications to mesh generation. Int J Numer Methods Eng 7:461–477MathSciNetCrossRefMATH
19.
Johnen A, Remacle JF, Geuzaine C (2011) Geometrical validity of curvilinear finite elements. In: 20th International meshing roundtable, Paris, pp 255-271
20.
Johnen A, Remacle JF, Geuzaine C (2014) Geometrical validity of high-order triangular finite elements. Eng Comput 30:375–382CrossRef
21.
22.
Sahni O, Xuo XJ, Janse KE, Shephard MS (2010) Curved boundary layer meshing for adaptive viscous flow simulations. FEAD 46:132–139MathSciNet
23.
Sherwin SJ, Peiro J (2002) Mesh generation in curvilinear domains using high-order elements. Int J Numer Method Eng 55:207–223CrossRefMATH
24.
Xuo XJ, Shephard MS, O’Bara RM, Natasia R, Beal MW (2004) Automatic p-version mesh generation for curved domains. Eng Comput 20:273–285CrossRef
Metadata
Title
Geometric validity (positive jacobian) of high-order Lagrange finite elements, theory and practical guidance
Authors
P. L. George
H. Borouchaki
N. Barral
Publication date
01-07-2016
Publisher
Springer London
Published in
Engineering with Computers / Issue 3/2016
Print ISSN: 0177-0667
Electronic ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-015-0422-1

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