Transformation Parameterization

A 3D Fourier series representation[17] is used to parameterize the forward and reverse transformations. This parameterization is simpler than the parameterizations used in our previous work [14,28,29] and each basis coefficient can be interpreted as the weight of a harmonic component in a single coordinate direction. Let $k=[k_1,k_2,k_3]$ and $n=[n_1,n_2,n_3]$. The displacement fields are defined to have the form

$$u_c (x) = \sum_{k \in \Omega_d} \mu[k] e^{j<x,N \theta [k]>} \quad w_c (x) = \sum_{k \in \Omega_d} \eta[k] e^{j<x,N \theta [k]>}$$ (6)

for $x \in \Omega_c$ where $\mu[k]$ and $\eta[k]$ are $(3 \times 1)$ complex-valued vectors, $\theta[k] = [\frac {2\pi k_1} {N_1}, \frac {2\pi k_2} {N_2}, \frac {2\pi k_3} {N_3}]^T$ and $N \theta [k] = [2\pi k_1, 2\pi k_2, 2\pi k_3]^T$. The notation $<\cdot, \cdot>$ denotes the dot product of two vectors such that $<x,N \theta[k]> = 2 \pi k_1 x_1 + 2 \pi k_2 x_2 + 2 \pi k_3 x_3$. The discretized displacement fields are defined as

$$u_d [n] = u_c (\frac n N) = \sum_{k \in \Omega_d} \mu[k] e^{j<n,\theta [k]>} \quad w_d [n] = w_c (\frac n N) = \sum_{k \in \Omega_d} \eta[k] e^{j<n,\theta [k]>}$$ (7)

for $n \in \Omega_d$. The basis coefficients are defined as $\mu [k] = \frac 1 {N_1 N_2 N_3} \sum_{n \in \Omega_d} u_d[n] e^{-j<n,\theta [k]>}$ and $\eta [k] = \frac 1 {N_1 N_2 N_3} \sum_{n \in \Omega_d} w_d[n] e^{-j<n,\theta [k]>}$ for $k \in \Omega_d$. The displacement fields associated with the inverse of the forward and reverse transformations are given by replacing $u$, $w$, $\mu$, and $\eta$ in Eq. 7 with $\tilde{u}$, $\tilde{w}$, $\tilde{\mu}$, and $\tilde{\eta}$, respectively.

The Fourier series parameterization is periodic in $x$ and therefore has cyclic boundary conditions for $x$ on the boundary of $\Omega$. Further more, the following proposition shows that the displacement fields are real assuming that the coefficients $\mu[k]$ and $\eta[k]$ have complex conjugate symmetry. It is shown in Section 2.6 that $\mu[k]$ and $\eta[k]$ have complex conjugate symmetry by construction.

Proposition 1   A displacement field of the form $u_d [n] = \sum_{k \in \Omega_d} \mu[k] e^{j<n,\theta [k]>}$, $n \in \Omega_d$ is real and can be written as

$$u_d [n] = 2 \sum_{k_1=0}^{(N_1/2)-1} \sum_{k_2=0}^{N_2-1} \sum_{k... ...^{j<n,\theta[k]>}\} - b[k] Im\{e^{j<n,\theta[k]>}\} \Big), \quad n \in \Omega_d$$ (8)

if the $(3 \times 1)$ vector $\mu[k] = a[k] + jb[k]$ has complex conjugate symmetry¹.

Proof
The displacement field $u_d$ can be written as

$$u_d [n] = \sum_{k \in \Omega_d} (a[k]+jb[k]) e^{j<n,\theta[k]>} ... ...}^{N_3-1} (a[k]+jb[k]) e^{j<n,\theta[k]>} + (a[k]-jb[k]) e^{-j<n,\theta[k]>}$$

because the $\mu[k]$ coefficients are complex conjugate symmetric, i.e., $\mu[k] = \mu^*[N-k]$. Simplifying the summand gives the result. Q.E.D.

The Fourier series parameterization in Eq. 6 is useful for simplifying the linear elasticity constraint given in Eq. 5. The operator $L_c$ can be thought of as a $(3\times 3)$ matrix differential operator[29] such that the linear elasticity operator $L_c = - \alpha \nabla^2 - \beta \nabla \cdot \nabla +\gamma = \begin{bmatrix} ... ... \nabla^2 - \beta \frac {\partial^2}{\partial x_3^2} + \gamma \end{bmatrix} .$ Substituting Eq. 7 into Eq. 5 and discretizing²the continuous partial derivatives of $L_c$ gives

$$C_{\text{REG}}(u) + C_{\text{REG}}(w) = \sum_{k \in \Omega_d} \mu^{\dag }[k] D^2[k] \mu[k] + \eta^{\dag }[k] D^2[k] \eta[k]$$ (9)

where $\dag$ is the complex conjugate transpose. $D[k]$ is a real-valued, $(3\times 3)$ matrix with elements

$$\Big[ D[k] \Big]_{rs} = \begin{cases} 2\alpha \Big\{ N_1^2 \big(1... ...big(\theta_r [k] \big) \sin \big(\theta_s [k] \big) , & r \neq s . \end{cases}$$

Likewise, the inverse consistency constraint Eq. 3 can be simplified in a similar manner. Substituting Eq. 7 into Eq. 4 and discretizing gives

$$C_{\text{ICC}}(u,\tilde{w}) + C_{\text{ICC}}(w,\tilde{u}) = \sum_... ...ilde{\eta}[k]) +(\eta[k] - \tilde{\mu}[k])^{\dag } (\eta[k] - \tilde{\mu}[k]) .$$ (10)

Copyright © 2002 • The University of Iowa. All rights reserved. Iowa City, Iowa 52242
Questions or Comments: gary-christensen@uiowa.edu