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

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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.

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Notes

  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.

Abbreviations

\({\hat{K}}\) :

The reference element, K the current element, \(F_K\) the mapping from \({\hat{K}}\) to K, \(p_i, p_{ij}, \ldots ,\) a shape function, d the degree of the finite element, \({\mathcal{J}}\) the jacobian of K, q the degree of this jacobian,

\({\hat{A}_i}\), \(A_i, (A_{ij}, A_{ijk}, A_{ijkl})\) :

A node of \({\hat{K}}\) and its image by \(F_K\)

u, v, w, t or \(\hat{x}, \hat{y}, \hat{z}\) :

The parameters living in the parametric space, e.g. \(\hat{K}\)

\(\varGamma\) and \(\gamma\), \(\Sigma\) and \(\sigma\), \(\Theta\) and \(\theta\), resp:

A curve and its expression, a bidimensional patch and its expression, a tridimensional patch and its expression,

\(P_{ij}\) (\(P_{ijk}, P_{ijkl}\)):

A control point, \(N_{ij}\) (\(N_{ijk}, N_{ijkl}\)) a (scalar) control value

\(B^d_{i} (u)\) :

The Bernstein polynomial of degree d for a system of natural coordinates, \(B^d_{ij} (u,v), B^d_{ijk}(u,v,w),\ldots ,\) the Bernstein polynomial of degree d for a system of barycentric coordinates, \(C^d_{i}\), the binomial coefficient,

\(\left[ .\right]\), \(\left\{ .\right\}\),\(\left| .\right|\), \((.\wedge .)\) and \(<.>\) :

A matrix, a vector, a determinant, the cross product and the dot product.

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Authors and Affiliations

  1. INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153, Le Chesnay Cedex, France

    P. L. George, H. Borouchaki & N. Barral

  2. UTT, ICD-Gamma3, Université de Technologie de Troyes, CS 42060, 10004, Troyes Cedex, France

    H. Borouchaki

Authors
  1. P. L. George
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  2. H. Borouchaki
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  3. N. Barral
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Correspondence to P. L. George.

Appendix

Appendix

1.1 Edge nodes versus control points

In this Appendix, we give some tables where it is shown, for the first orders (from 1 to 5), how the edge nodes are related to the edge control points and, conversely, how the edge control points can be found when the data are made up of the edge nodes (as it is in the finite element world).

Tables 7 and 8 depict the case of curved edges where natural indices are used (as it is for quads, quadrilateral faces, hexes, ...). Tables 9 and 10 are identical but here we used the indices as they are defined in a barycentric system (as it is for triangle, triangular faces and tetrahedra), and therefore, we only give the degree 3. The relation from one system to the other is as follows (here for the degree 3):

figure c

Since complete elements and reduced elements have the same boundary edges, the relations hold for both cases.

In the tables, we consider the case of bidimensional edges but, obviously, this apply to three dimensions after an adequate labelling of the entities.

Table 7 Complete or Serendipity Lagrange quadrilaterals of degree 2 to 5
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Table 8 Complete or Serendipity Lagrange quadrilaterals of degree 2 to 5
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Table 9 Complete or reduced Lagrange triangle of degree 3
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Table 10 Complete or reduced Lagrange triangle of degree 3
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1.2 Determinants in a control coefficient

We explain how to find the determinants involved in one control coefficient by considering the case of a quad of degree d, Relation (27).

At first, it is required to find the appropriate indices \(i_1\) and \(i_2\) such that (\(q=2d-1\))

  • (For \(I=0,q\), (for \(i_1=0,d-1\) (for \(i_2 = 0,d\) (\(i_1+i_2 = I\))))),

from which we have the pairs \((i_1,i_2)\) therefore the pertinent vectors together with the associated weights (the factors \(C^{..}_{..}\)) relative to these indices. Then we do the same for the indices \(j_1\) and \(j_2\) so we find the pairs such that

  • ( For \(J=0,q\), (for \(j_1=0,d\) (for \(j_2 = 0,d-1\) (\(j_1+j_2 = J\))))),

so we have the pairs \((j_1,j_2)\) therefore the pertinent vectors together with the associated weights (the factors \(C^{..}_{..}\)) relative to these indices. Now, we group together these indices to obtain the pairs \((i_1,j_1)\) and \((i_2,j_2)\), the weights and the number of terms (determinants) in a given coefficient. In Table 11 and the following schematics, we give details about the coefficients \(N_{IJ}\) when \(J=0\), note that \(vector_1\) span the edge \(P_{00}P_{d0}\), in the u-direction, from \(\overrightarrow{P_{00}P_{10}}\) to \(\overrightarrow{P_{I0}P_{I+1,0}}\) (top to bottom) while \(vector_2\) are the vectors in the v-direction from \(\overrightarrow{P_{00}P_{01}}\) to \(\overrightarrow{P_{I0}P_{I1}}\) (bottom to top). For \(J=1\), we have to repeat the same task but now we have two pairs in \((j_1,j_2)\), one being (0, 1), and the other being (1, 0), Table 12, increasing the number of determinants in a given coefficient, etc. Finding all the coefficients clearly requires writing a computer program.

Table 11 Example of how to compute some control coefficients when \(J=0\)
Full size table
figure d
figure e
Table 12 Example of how to compute some control coefficients when \(J=1\)
Full size table

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George, P.L., Borouchaki, H. & Barral, N. Geometric validity (positive jacobian) of high-order Lagrange finite elements, theory and practical guidance. Engineering with Computers 32, 405–424 (2016). https://doi.org/10.1007/s00366-015-0422-1

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