Page 181 - Advances in Biomechanics and Tissue Regeneration

REFERENCES 177

particles with a nonvanishing weight function at point θ constitutes its support denoted by Λ. Based on this set of par-
ticles, a weighted least square fit in the vicinity of point θ can be constructed according to

θ θ
I
X 2
½Pðθ aðθÞ fðθ Þފ Φ : (A.2)
I
I
JðaðθÞÞ :¼ ϱ
I2Λ I
The unknown coefficients a(θ) can be readily determined by minimizing the squared and weighted error between the
I
I
approximated and actual values of f(θ ) at each of the particles θ 2 Λ. That is, minimizing the function J (Eq. A.2), with
respect to a(θ). Finally, the coefficients a(θ) are substituted into Eq. (A.1) and the approximation of function f(θ) takes
the following form:
θ θ
I
X
I
h 1 Pðθ ÞΦ f I ,
f ðθÞ¼ PðθÞ M ðθÞ ϱ (A.3)
I2Λ I
where M(θ) is the so-called moment matrix of the weight function Φ,
θ θ
I
X I I
Pðθ ÞPðθ ÞΦ ,
MðθÞ¼ ϱ (A.4)
I2Λ I
and f I are the so-called particle parameters.
Clearly, as the least square fit, Eq. (A.2) only includes a very limited number of sample points, that is, the particles
I
h
θ 2 Λ, the local character of the approximation f (θ) is ensured. With the refinement of the particle distribution the
h
support of the weight functions can be chosen smaller and the approximation f (θ) converges for ϱ ! 0 to the exact
I
function f(θ). The minimum number of supporting particles for any point θ is dictated by the invertibility requirement
of Eq. (A.4), that is, the chosen basis polynomial. The smoothness of the MLS approximation (Eq. A.3) depends on the
m l h k
continuity of the basis polynomial P 2 C (Ω) as well as the weight function Φ 2 C (Ω) and it holds f 2 C with
k ¼ minðl,mÞ [33].
The chosen complete linear basis polynomial in an n-dimension is given by
1
n
PðθÞ¼½1,θ ,…,θ Š, (A.5)
and the weight function is based on a cubic spline:
2 1
8
4r +4r
> 2 3
>
> for jrj
> 3 2
>
4 2 4 3 1
<
4r +4r r for jrj 1: (A.6)
wðrÞ¼
3 3 2
>
>
> 0 for jrj > 1
>
>
:
For the two-dimensional domain the weight function can be constructed by the following expression:
θ θ θ 1 θ θ 2 θ θ n θ
I I I I
Φ ¼ w 1 ,…,w 2 ,…,w n : (A.7)
ϱ I ϱ I ϱ I ϱ I
Acknowledgments
This research has been supported by the Center for High Performance Computing South Africa and the National Research Foundation of South
Africa (Grant Numbers 90528, 93111, 104839, and 105858). Opinions expressed and conclusions arrived at are those of the authors and are not nec-
essarily to be attributed to the NRF.
References
[1] L.L. Demer, F.C.P. Yin, Passive biaxial mechanical properties of isolated canine myocardium, J. Physiol. 339 (1983) 615–630.
[2] J.M. Guccione, L.K. Waldman, A.D. McCulloch, Mechanics of active contraction in cardiac muscle: part II—cylindrical models of the systolic left
ventricle, J. Biomech. Eng. 115 (1) (1993) 82–90.
[3] P.C. Franzone, L. Guerri, Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I. Validation of the Eikonal model, Math. Biosci.
113 (2) (1993) 145–209.

I. BIOMECHANICS