Measures describing curvilinear short fiber distributions

Hartmann, Stefan; Liese, Teresa

doi:10.1007/s00419-020-01762-8

Cited by 2 publications

(5 citation statements)

References 9 publications

(12 reference statements)

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“…There are n (j) px pixel of each line j = 1, 2. Since there is interest in interpolating the points to obtain a smooth representation, third-order splines are drawn on as discussed in [3,8]. There are several possibilities to handle the data: first, each pixel is treated as a measurement point.…”

Section: Problem Descriptionmentioning

confidence: 99%

“…Since some of the tools developed later on are also applicable to three-dimensional problems, the vectors might be elements of V 3 . In analogy to [8], piece-wise polynomials of third-order are drawn on as an interpolation concept…”

Section: Interpolation Conceptmentioning

confidence: 99%

“…There are

n_{px}^{(j)}

pixel of each line

j = 1, 2

. Since there is interest in interpolating the points to obtain a smooth representation, third‐order splines are drawn on as discussed in [3,8].…”

Section: Geometrical Considerationsmentioning

confidence: 99%

“…In analogy to [8], piece‐wise polynomials of third‐order are drawn on as an interpolation concept

{\hat{x}}_{k i}^{(j)} (ξ^{(j)}) = a_{k i}^{(j)} + b_{k i}^{(j)} (ξ^{(j)} - ξ_{i}^{(j)}) + c_{k i}^{(j)} {(ξ^{(j)} - ξ_{i}^{(j)})}^{2} + d_{k i}^{(j)} {(ξ^{(j)} - ξ_{i}^{(j)})}^{3},

for

ξ_{i}^{(j)} \leq ξ^{(j)} < ξ_{i + 1}^{(j)}

i = 1, \dots, n_{d}^{(j)} - 1

. The values

ξ_{i}^{(j)}

are so‐called sampling points which are determined by the Chord‐approach (here, a normalization is assumed)

{\hat{ξ}}_{i}^{<...}

…”

Section: Geometrical Considerationsmentioning

confidence: 99%

“…Thus, the distribution of the thickness and also the calculation of the thickness change can be provided semi‐automatically. The basis is a method for the geometrical representation of short fibers in composites, which is applied here and extended to the special application under consideration, see [8]. From an image, we determine the curves (contour lines) between the layers.…”

Section: Introductionmentioning

confidence: 99%

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Thickness determination of laminate core layers from microscopy images

et al. 2022

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In sandwich laminates made of metal–polymer–metal (MPM) layers under forming conditions, the thickness changes along the thin‐layered specimens are of particular interest. To measure this distribution, it is common to take microscopic images and determine manually the thicknesses at particular discrete points. We compare and provide several methods determining the thickness distribution from microscopic images. First, an interpolation scheme is chosen to obtain a continuous function of the interfaces between the layers instead of the discrete pixel data. Afterwards, the methods—ruled surface approach, orthogonal projection, embedded circle scheme, and a dull but simple method—are applied and compared with each other. For very smooth data, most of the schemes show equivalent results. However, the “brute force”—called the dull approach—turns out to be the most robust scheme, particularly, if noisy data is studied. The Gaussian error propagation concept is applied to study the uncertainties resulting from noisy pixel data. The schemes are adapted to finite element simulation results as well so that a direct comparison of experimental and numerical data would be possible.

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Section: Problem Descriptionmentioning

confidence: 99%

Section: Interpolation Conceptmentioning

confidence: 99%

“…There are