Automatic 3D Finite Element Mesh Generation : data fitting from an atlas
Luboz Vincent*, Payan Yohan*, Swider Pascal**, Couteau Béatrice**
*TIMC-IMAG Laboratory, GMCAO **INSERM, U518
Summary
Finite Element (FE) analysis starts to be largely used in the field of computer assisted surgery, from bones model in orthopaedic surgery to facial tissues modelling in craniofacial
surgery. In those frameworks, models have to be rebuilt and adapted to each patient, which means the use of automatic 3D mesh generators. In practice, to maintain a strong patientoriented specificity, such automatic generators often make some compromises in terms of homogeneity, symmetry or mesh refinement. For those reasons, a new mesh generating
technique, namely the Mesh-Matching algorithm ([1], [2]), has been introduced. Five steps need to be driven to generate a patient FE model: (1) the atlas generation, (2) the
segmentation of the patient data, (3) the computation (following a matching process) of the transformation T between the atlas and the patient data, (4) the application of T to the atlas to
generate a patient mesh and (5) the regularization (based on the computation of the jacobian determinant) of the patient mesh to allow FE analysis.
(1) & (2) Definition of the generic FE model: the “ atlas ” :
(5) Post-treatment of the mesh irregularities :
An element is irregular if the jacobian matrix is singular.
(a) Segmentation of CT scan data
to get the external surface of the
proximal femora.
(b) Manual definition of the atlas
FE mesh
(b)
(a)
(3) & (4) Automatic generation of a patient mesh :
In 2D : dA = dx dy = detJ dr ds
(a) Computation of the elastic transformation T between
atlas points (from the external surface of the atlas mesh ; in
blue) and patient point (from the segmentation step ; in
pink), according to the Mesh Matching method [1].
(b) Application of T to every nodes of the atlas mesh (i.e.
internal and external nodes) to generate the patient mesh.
(a)
Results:
Evaluation on 10 proximal femora and 5 entire femora [2] :
each generated mesh allows FE analysis.
Tests on others biomechanical structures (for example,
human face model) : regularity problems.
Post-treatment required.
(b)
In 3D : dV = dx dy dz = detJ dr ds dt
Regularity criterion : detJ must be strictly positive at each node of the element.
Regularization process :
(1) Computation of the jacobian determinant value detJ for each node of the mesh.
(2) Detection of irregular elements (nodes with a negative or null detJ value)
(3) Correction of each irregular elements by moving nodes in the direction of the gradient of detJ:
n(i+1) = n(i) + h * detJ(i)
where n(i) and n(i+1) are the coordinates of the node n at step i and i+1, h is the
factor of convergence detJ(i) is the gradient of detJ computed at node i.
Result of the regularization on a very simple mesh
(a)
Position of the internal node in function of min
detJ
1,2
Distance (mm)
1
0,8
0,6
0,4
0,2
0
0
0,02 0,04 0,06 0,08
min detJ
Result of the regularization on a complex mesh
(c)
(b)
-0,12 -0,1 -0,08 -0,06 -0,04 -0,02
with dA area of an element
<x, y> actual element coordinates
<r, s> reference element coordinates.
0,1
0,12
(a) Cubic regular mesh (2 mm edge length):
min detJ = 0.125, the internal node
position is ”perfect”.
(b) Irregular mesh created by moving the
internal node : min detJ = -0.1125, its
internal node is located at 1.17 mm of the
“perfect position” (of the cubic mesh).
(c) Corrected element after the posttreatment : with detJ > 0.02), its internal
node is located at 0.48 mm of its “perfect
position”.
improvement of the mesh quality.
(b)
(c)
(a)
(a) Mesh generated by the M-M method: 149 irregular elements (with detJ < 0) on 2884.
(b) Example of an irregular element of the mesh (twisted during the M-M application).
(c) Element regularized by the post-treatment
Finite Element analysis is now allowed
References
[1] Couteau B., Payan Y. and Lavallée S. (2000). The Mesh-Matching algorithm : an automatic 3D mesh generator for finite element structures, Journal of Biomechanics, vol. 33/8, 1005-1009.
[2] Luboz V., Couteau B., Payan Y., 2001. 3D Finite element meshing of entire femora by using the mesh matching algorithm. In Proceedings of the Transactions of the 47th Annual Meeting of the Orthopaedic Research
Society, p. 522. Transactions Editor, B. Stephen.