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Research Papers

Developing Metamodels for Fast and Accurate Prediction of the Draping of Physical Surfaces

[+] Author and Article Information
Esben Toke Christensen

Department of Mechanical and
Manufacturing Engineering,
Aalborg University,
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: esben@m-tech.aau.dk

Alexander I. J. Forrester

Centre of Excellence,
University of Southampton Engineering,
Burgess Road,
Southampton SO16 7QF, UK
e-mail: alexander.forrester@soton.ac.uk

Erik Lund

Department of Mechanical and
Manufacturing Engineering,
Aalborg University
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: el@m-tech.aau.dk

Esben Lindgaard

Department of Mechanical and
Manufacturing Engineering,
Aalborg University,
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: elo@m-tech.aau.dk

Manuscript received April 18, 2017; final manuscript received October 28, 2017; published online March 15, 2018. Assoc. Editor: Ying Liu.

J. Comput. Inf. Sci. Eng 18(2), 021003 (Mar 15, 2018) (12 pages) Paper No: JCISE-17-1080; doi: 10.1115/1.4039334 History: Received April 18, 2017; Revised October 28, 2017

In this paper, the use of methods from the meta- or surrogate modeling literature, for building models predicting the draping of physical surfaces, is examined. An example application concerning modeling of the behavior of a variable shape mold is treated. Four different methods are considered for this problem. The proposed methods are difference methods assembled from the methods kriging and proper orthogonal decomposition (POD) together with a spline-based underlying model (UM) and a novel patchwise modeling scheme. The four models, namely kriging and POD with kriging of the coefficients in global and local variants, are compared in terms of accuracy and numerical efficiency on data sets of different sizes for the treated application. It is shown that the POD-based methods are vastly superior to models based on kriging alone, and that the use of a difference model structure is advantageous. It is demonstrated that patchwise modeling schemes, where the complete surface behavior is modeled by a collection of locally defined smaller models, can provide a good compromise between achieving good model accuracy and scalability of the models to large systems.

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Figures

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Fig. 1

Illustration of the variable shape mold concept

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Fig. 2

Layout of the substitute model. Each dot represents an actuator. Dimensions as well as the selected actuator numbering are shown.

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Fig. 3

Schematic illustration of the actuator to membrane implementation in the numerical substitute model

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Fig. 4

Illustration of the interpolation done during post-processing. Due to in-plane displacements of the FE nodes, a piecewise linear interpolation is used to evaluate the displacements in a fixed grid to achieve a consistent data representation.

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Fig. 5

The principle in the difference method in two cases: (a) at training and (b) at prediction. The punctuated box encloses what is collectively referred to as the surrogate model.

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Fig. 6

Illustration of the construction of the spline-based coarse model. Splines are fitted and sampled along the in-plane directions in the succession shown in the figure legend. The total number of sampled points in the plane is N2 (see Table 1).

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Fig. 7

Illustration of the arrangement of patch models on the mold surface. Translucent squares mark the prediction regions of three models. The red square in full line marks the input region of the red patch model. Squares in broken black line exemplify quarter, half, and full size patch models.

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Fig. 8

Comparison of the UM and the four surrogate models. Mean, standard deviation, and min/max are shown for the RMAE (Eq. (35)) and RRMSE (Eq. (36)) measures. Numbers attached to each point on the mean graphs indicate the number of mold configurations in the data set for that particular model.

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Fig. 9

Plots of the hyper values for krigs trained on 9 × 9 spatial domains. The hyper parameters shown correspond to (a) actuator 1 with center (0,0), (b) actuator 41 with center (0.5 L,0.5L) (c) actuator 72 with center (L,0), and (d) the x coordinate.

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Fig. 10

Comparison of the UM and the four approximation methods without the use of an UM. Mean, standard deviation, and min/max are shown for the RMAE (Eq. (35)) and RRMSE (Eq. (36)) measures. Numbers attached to each point on the mean graphs indicate the number of mold configurations in the data set for that particular model. It is seen that without including the knowledge of an approximate UM, the models struggle to even reach the base performance of the UM.

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