Assessing the Use of Tensor Closure Methods With Orientation Distribution Reconstruction Functions

Higher Order ◽

Interaction Coefficients ◽

Order Tensor ◽

Orientation Tensor ◽

Orientation Tensors

Orientation tensors are widely used to describe fiber distri-butions in short fiber reinforced composite systems. Although these tensors capture the stochastic nature of concentrated fiber suspensions in a compact form, the evolution equation for each lower order tensor is a function of the next higher order tensor. Flow calculations typically employ a closure that approximates the fourth-order orientation tensor as a function of the second order orientation tensor. Recent work has been done with eigen-value based and invariant based closure approximations of the fourth-order tensor. The effect of using lower order tensors tensors in process simulations by reconstructing the distribution function from successively higher order orientation tensors in a Fourier series representation is considered. This analysis uses the property that orientation tensors are related to the series expansion coefficients of the distribution function. Errors for several closures are investigated and compared with errors developed when using a reconstruction from the exact 2nd, 4th, and 6th order orientation tensors over a range of interaction coefficients from 10−4 to 10−1 for several flow fields.

A Statistical Method to Obtain Material Properties From the Orientation Distribution Function for Short-Fiber Polymer Composites

Materials ◽

10.1115/imece2005-81263 ◽

2005 ◽

Author(s):

David A. Jack ◽

Douglas E. Smith

Keyword(s):

Distribution Function ◽

Material Properties ◽

Fiber Orientation ◽

Short Fiber ◽

Expectation Values ◽

Order Tensor ◽

Fiber Orientation Distribution ◽

Orientation Tensors

Material behavior of short-fiber composites can be found from the fiber orientation distribution function, with the only widely accepted procedure derived from the application of orientation/moment tensors. The use of orientation tensors requires a closure, whereby the higher order tensor is approximated as a function of the lower order tensor thereby introducing additional computational errors. We present material property expectation values computed directly from the fiber orientation distribution function, thereby alleviating the closure problem inherent to orientation tensors. Material properties are computed from statistically independent unidirectional fiber samples taken from the fiber orientation distribution function. The statistical nature of the distribution function is evaluated with Monte-Carlo simulations to obtain approximate stiffness tensors from the underlying unidirectional composite properties. Examples are presented for simple analytical distributions to demonstrate the effectiveness of expectation values and results are compared to properties obtained through orientation tensors. Results yield a value less than 1.5% for the coefficient of variation and suggest that the orientation tensor method for computing material properties is applicable only for the case of non-interacting fibers.

Fast Solutions for the Fiber Orientation of Concentrated Suspensions of Short-Fiber Composites Using the Exact Closure Method

Volume 3: Design and Manufacturing, Parts A and B ◽

10.1115/imece2010-39479 ◽

2010 ◽

Author(s):

Stephen Montgomery-Smith ◽

David A. Jack ◽

Douglas Smith

Keyword(s):

Distribution Function ◽

Fiber Orientation ◽

Equation Of Motion ◽

Fiber Composites ◽

Short Fiber ◽

Second Order ◽

Orientation Tensor ◽

Kinetics Of

The kinetics of the fiber orientation during processing of short-fiber composites governs both the processing characteristics and the cured part performance. The flow kinetics of the polymer melt dictates the fiber orientation kinetics, and in turn the underlying fiber orientation dictates the bulk flow characteristics. It is beyond computational comprehension to model the equation of motion of the full fiber orientation probability distribution function. Instead, typical industrial simulations rely on the computationally efficient equation of motion of the second-order orientation tensor (also known as the second-order moment of the orientation distribution function) to model the characteristics of the fiber orientation within a polymer suspension. Unfortunately, typical implementation forms of any order orientation tensor equation of motion requires the next higher, even ordered, orientation tensor, thus necessitating a closure of the higher order expression. The recently developed Fast Exact Closure avoids the classical closure problem by solving a set of related second-order tensor equations of motion, and yields the exact solution for pure Jeffery’s motion as the diffusion goes to zero. Typical closures are obtained through a fitting process, and are often obtained by fitting for orientation states obtained from solutions of the full orientation distribution function, thus tying the closure to the flows from which it was fit. With the recent understandings of the limitations of the Folgar and Tucker (1984) model of fiber interactions during processing, it has become clear the importance of developing a closure that is independent of any choice of fitting data. The Fast Exact Closure presents an alternative in that it is constructed independent of any fitting process. Results demonstrate that when diffusion exists, the solution is not only physical, but solutions for flows experiencing Folgar-Tucker diffusion are shown to exhibit an equal to or greater accuracy than solutions relying on closures developed via a curve fitting approach.

Sparse deconvolution of higher order tensor for fiber orientation distribution estimation

Artificial Intelligence in Medicine ◽

10.1016/j.artmed.2015.09.004 ◽

2015 ◽

Vol 65 (3) ◽

pp. 229-238 ◽

Cited By ~ 5

Author(s):

Yuanjing Feng ◽

Ye Wu ◽

Yogesh Rathi ◽

Carl-Fredrik Westin

Keyword(s):

Fiber Orientation ◽

Higher Order ◽

Order Tensor ◽

Distribution Estimation ◽

Fiber Orientation Distribution

Acoustoelastic determination of the higher order orientation distribution function coefficients up tol = 12 and their use for the on-line prediction ofr-value

Metallurgical Transactions A ◽

10.1007/bf02671940 ◽

1990 ◽

Vol 21 (2) ◽

pp. 697-706 ◽

Cited By ~ 13

Author(s):

K. Sakata ◽

D. Daniel ◽

J. J. Jonas ◽

J. F. Bussiére

Keyword(s):

Distribution Function ◽

Higher Order ◽

On Line

Calculation of Fourth-Order Tensor Product Expansion by Power Method and Comparison of it with Higher-Order Singular Value Decomposition

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.444-445.703 ◽

2013 ◽

Vol 444-445 ◽

pp. 703-711

Author(s):

Akio Ishida ◽

Takumi Noda ◽

Jun Murakami ◽

Naoki Yamamoto ◽

Chiharu Okuma

Keyword(s):

Singular Value Decomposition ◽

Tensor Product ◽

Fourth Order ◽

Computation Time ◽

Higher Order ◽

Singular Value ◽

Multidimensional Data ◽

Power Method ◽

Order Tensor ◽

Value Decomposition

Higher-order singular value decomposition (HOSVD) is known as an effective technique to reduce the dimension of multidimensional data. We have proposed a method to perform third-order tensor product expansion (3OTPE) by using the power method for the same purpose as HOSVD, and showed that our method had a better accuracy property than HOSVD, and furthermore, required fewer computation time than that. Since our method could not be applied to the tensors of fourth-order (or more) in spite of having those useful properties, we extend our algorithm of 3OTPE calculation to forth-order tensors in this paper. The results of newly developed method are compared to those obtained by HOSVD. We show that the results follow the same trend as the case of 3OTPE.

A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation

Geophysics ◽

10.1190/geo2011-0399.1 ◽

2012 ◽

Vol 77 (3) ◽

pp. V113-V122 ◽

Cited By ~ 96

Author(s):

Nadia Kreimer ◽

Mauricio D. Sacchi

Keyword(s):

Fourth Order ◽

Seismic Data ◽

Temporal Frequency ◽

Reduction Process ◽

Higher Order ◽

Singular Value ◽

Low Rank ◽

High Signal ◽

Order Tensor ◽

Fourth Order Tensor

A patch of prestack data depends on four spatial dimensions ([Formula: see text], [Formula: see text] midpoints and [Formula: see text], [Formula: see text] offsets) and frequency. The spatial data at one temporal frequency can be represented by a fourth-order tensor. In ideal conditions of high signal-to-noise ratio and complete sampling, one can assume that the seismic data can be approximated via a low-rank fourth-order tensor. Missing samples were recovered by reinserting data obtained by approximating the original noisy and incomplete data volume with new observations obtained via the rank-reduction process. The higher-order singular value decompostion was used to reduce the rank of the prestack seismic tensor. Synthetic data demonstrated the ability of the proposed seismic data completion algorithm to reconstruct events with curvature. The synthetic example allowed to quantify the quality of the reconstruction for different levels of noise and survey sparsity. We also provided a real data example from the Western Canadian sedimentary basin.

Earing in Deep-Drawing Steels

Texture ◽

10.1155/tsm.1.173 ◽

1974 ◽

Vol 1 (3) ◽

pp. 173-182 ◽

Cited By ~ 7

Author(s):

G. J. Davies ◽

D. J. Goodwill ◽

J. S. Kallend

Keyword(s):

Distribution Function ◽

Cold Rolling ◽

Fourth Order ◽

Deep Drawing ◽

Rolling Reduction ◽

Crystallite Orientation ◽

Cold Rolling And Annealing ◽

Cold Rolled

The variations of the fourth-order coefficients of the crystallite orientation distribution function, with rolling reduction have been determined after cold-rolling and annealing for a deep-drawing quality rimming steel and an aluminium-killed steel. These coefficients influence drawability and earing behaviour and by the manipulation of the coefficients in the distribution function of a 60% cold-rolled and annealed rimming steel, a hypothetical non-earing sheet texture has been derived. By comparison with the actual sheet texture those textural components which most affect earing behaviour are identified.