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Research Papers

Fidelity of the Estimation of the Deformation Gradient From Data Deduced From the Motion of Markers Placed on a Body That is Subject to an Inhomogeneous Deformation Field

[+] Author and Article Information
Vít Průša

Faculty of Mathematics and Physics,
Charles University in Prague,
Sokolovská 83,
Praha, CZ 18675, Czech Republic
e-mail: prusv@karlin.mff.cuni.cz

K. R. Rajagopal

Texas A&M University,
Department of Mechanical Engineering,
3123 TAMU,
College Station, TX 77843-3123
e-mail: krajagopal@tamu.edu

U. Saravanan

Department of Civil Engineering,
Chennai 600036, India
e-mail: saran@iitm.ac.in

Saravanan et al. [14] reported that the deformation of circumflex coronary arteries does not conform to the expected deformation field. The same has been confirmed for other arteries (see Refs. [15] and [16]) also.

The procedure for obtaining the estimate of the gradient by tracking the positions of the markers is interpreted, as it is common in the experimental mechanics community. (First, a linear approximation of the deformation field is found, and then the estimate of the deformation gradient is obtained by finding the deformation gradient of the approximated deformation field.) A mathematically inclined person would rather say that the gradient (“the tangent”) at a given point is estimated by a linear fit (“a chord”) of the deformation field through some adjoining points.

In what follows, we freely interpret $Lf(a)$ either as a functional or as a vector. This is acceptable in $ℝ3$ with the standard scalar product. Further, we use the same symbol $|·|$ for various norms—magnitude of a scalar, Euclidean norm of a vector, and the standard norm of a linear functional or linear operator. We recall that, if $F$ is a functional on vector space $V$, then the norm of the functional is defined as $|F|=supv∈V,v≠0|Fv||v|$. If $F$ is interpreted as a vector in $ℝ3$, then one can also find its Euclidean norm and it turns out that both norms are equivalent.

By

$L({w}i=1k)$
, we denote the space spanned by the vectors ${w}i=1k$.

Whenever we interpret an angle between two vectors as the minimal angle in the sense of Definition 2, we assume that the vectors are oriented in such a way that the angle takes values in $[0,π/2]$.

Note that $t$ that maximizes the second derivative is, in general, different for each component of $f$. This observation explains why, for a general vector valued function, it is impossible to get an equality of type 8 or 11.

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 7, 2012; final manuscript received December 28, 2012; accepted manuscript posted February 12, 2013; published online June 12, 2013. Assoc. Editor: Jeffrey W. Holmes.

J Biomech Eng 135(8), 081004 (Jun 12, 2013) (12 pages) Paper No: BIO-12-1397; doi: 10.1115/1.4023629 History: Received September 07, 2012; Revised December 28, 2012; Accepted February 12, 2013

Abstract

Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error.

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Figures

Fig. 1

Projection to a nonorthonormal basis

Fig. 2

Plot of the variation of 2.6 times the maximum value of the absolute value of the components of the second Fréchet derivative of the deformation given by Eq. (47) for different marker sets as a function of the deformed outer radius of the cylinder, ro. (The values predicted by set - 1, set – 2, and set - 4 are the same.)

Fig. 3

Normalized error evaluated at (Xm,Ym,Zm) in the predicted invariants (a) (J1c-J1e)/J1e, (b) (J2c-J2e)/J2e, and (c) (J3c-J3e)/J3e for different sets of four markers listed in Table 2 as a function of deformed outer radius, ro. (Set 1 and 3 predict the same value for J3c.)

Fig. 4

Plot of the variation of the distance between the estimated and actual deformed coordinates of the point (Xm,Ym,Zm), derr/Ro for different marker sets as a function of the deformed outer radius of the cylinder, ro

Fig. 5

Normalized error with regard to the invariants (a) (J1c-J1e)/J1e, (b) (J2c-J2e)/J2e, and (c) (J3c-J3e)/J3e for the linear deformation field determined using four markers in the set - 4 listed in Table 2 at (Xm,Ym,Zm) and at different marker locations. (Normalized error ɛ1 and ɛ2 when evaluated at locations corresponding to markers 1 and 6 and corresponding to markers 22 and 23 are the same. Normalized error ɛ3 is the same, irrespective of the location where it is evaluated.)

Fig. 6

Normalized error evaluated at (Xm,Ym,Zm) in the predicted invariants (a) (J1c-J1e)/J1e, (b) (J2c-J2e)/J2e, and (c) (J3c-J3e)/J3e for different sets of markers listed in Table 4 as a function of deformed outer radius, ro

Fig. 7

Squared error (SE) for different sets of markers listed in Table 4 as a function of deformed outer radius, ro

Fig. 9

Normalized error evaluated at (Xm,Ym,Zm) with regard to the invariant (J1c-J1e)/J1e for set - 7 markers listed in Table 5 for different approximations of the deformation fields as a function of deformed outer radius, ro

Fig. 8

Normalized error evaluated at (Xm,Ym,Zm) with regard to the invariants (a) (J1c-J1e)/J1e, (b) (J2c-J2e)/J2e, and (c) (J3c-J3e)/J3e for different sets of ten markers listed in Table 5 and different approximations of the deformation fields, as a function of deformed outer radius, ro

Fig. 10

Normalized error in the predicted invariants (a) (J1c-J1e)/J1e, (b) (J2c-J2e)/J2e, and (c) (J3c-J3e)/J3e for quadratic deformation field determined using ten markers in set - 7 listed in Table 5 at (Xm,Ym,Zm) and at different marker locations

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