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Research Papers: Variability/Uncertainty in D3

A Taylor Expansion Approach for Computing Structural Performance Variation From Population-Based Shape Data

[+] Author and Article Information
Xilu Wang

Computational Design & Manufacturing
Laboratory,
Department of Mechanical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705

Xiaoping Qian

Computational Design & Manufacturing
Laboratory,
Department of Mechanical Engineering,
University of Wisconsin-Madison,
Madison, WI 53705
e-mail: qian@engr.wisc.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 15, 2017; final manuscript received May 25, 2017; published online October 2, 2017. Assoc. Editor: Charlie C. L. Wang.

J. Mech. Des 139(11), 111411 (Oct 02, 2017) (11 pages) Paper No: MD-17-1134; doi: 10.1115/1.4037252 History: Received February 15, 2017; Revised May 25, 2017

Rapid advancement of sensor technologies and computing power has led to wide availability of massive population-based shape data. In this paper, we present a Taylor expansion-based method for computing structural performance variation over its shape population. The proposed method consists of four steps: (1) learning the shape parameters and their probabilistic distributions through the statistical shape modeling (SSM), (2) deriving analytical sensitivity of structural performance over shape parameter, (3) approximating the explicit function relationship between the finite element (FE) solution and the shape parameters through Taylor expansion, and (4) computing the performance variation by the explicit function relationship. To overcome the potential inaccuracy of Taylor expansion for highly nonlinear problems, a multipoint Taylor expansion technique is proposed, where the parameter space is partitioned into different regions and multiple Taylor expansions are locally conducted. It works especially well when combined with the dimensional reduction of the principal component analysis (PCA) in the statistical shape modeling. Numerical studies illustrate the accuracy and efficiency of this method.

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Topics: Shapes
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Figures

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Fig. 1

Proposed approach for predicting subject-specific structural performance: Taylor series expansion of the FE solution for the mean shape as applied in a heat transfer problem. (a) A shape population, (b) FE mesh of the mean shape, (c) FE result of (b), and (d) subject-specific temperature distributions from Taylor expansion.

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Fig. 2

Statistical shape modeling of hand shapes: (a) 40 aligned training shapes, (b) the mean shape: X¯Γ, (c) the first mode by varying w1: X¯Γ+w1ψ1, (d) the second mode by varying w2: X¯Γ+w2ψ2, and (e) the third mode by varying w3: X¯Γ+w3ψ3

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Fig. 3

Partition the domain into different regions and conduct Taylor expansion in each region separately: (a) three regions and (b) five regions

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Fig. 4

A 2D heat transfer problem: Dirichlet boundary condition u = 50 on Γ1 (the circles), Neumann boundary condition ∂u/∂n=−200 (the highlighted lines), thermal load: q = 1,000,000 in the center of the hand (the highlighted area). (a) Boundary conditions on the mean shape and (b) FE solution.

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Fig. 5

Predicted temperature distribution due to shape variations in the first mode. The darkness and brightness shows the temperature, and its range follows the same as that in Fig. 4. (a) ũ(w1)|w1=−2σ1, (b) ũ(w1)|w1=−σ1, (c) ũ(w1)|w1=σ1, and (d) ũ(w1)|w1=2σ1.

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Fig. 6

Comparing Taylor expansion with FE analysis of thermal compliance: (a) c̃=c(0)+w1∂c/∂w1, (b) c̃=c(0)+w2∂c/∂w2, (c) c̃=c(0)+w3∂c/∂w3, and (d) cumulative distribution functions from Taylor expansions and from 500 Monte Carlo simulations. (a) First mode, (b) second mode, (c) third mode, and (d) CDF.

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Fig. 7

The errors between the temperatures predicted by Taylor expansion and from FE analysis: (a) ũ(w1)−u(w1)|w1=−2σ1, (b) ũ(w1)−u(w1)|w1=2σ1, (c) ũ(w2)−u(w2)|w2=−2σ2, (d) ũ(w2)−u(w2)|w2=2σ2, (e) ũ(w3)−u(w3)|w3=−2σ3, and (f) ũ(w3)−u(w3)|w3=2σ3

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Fig. 8

A 2D linear elasticity problem: (a) FE model of the mean shape: Dirichlet boundary condition û=[0,0]T on ΓD (the circles), Neumann boundary condition t̂=[200,0]T (the highlighted lines) and (b) the resulting nodal displacements in horizontal. (a) Boundary conditions on the mean shape and (b) nodal displacements.

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Fig. 9

Taylor expansion predicted nodal displacements due to shape variations in the first mode. The darkness and brightness shows the values of horizontal displacements, and its range follows the same as that in Fig. 8. (a) ũ(w1)|w1=−2σ1, (b) ũ(w1)|w1=−σ1, (c) ũ(w1)|w1=σ1, and (d) ũ(w1)|w1=2σ1.

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Fig. 10

Comparing Taylor approximation with FE solutions of structural compliance: (a) first mode, (b) second mode, (c) third mode, and (d) CDF

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Fig. 11

Cumulative distribution function by multipoint Taylor expansion: (a) result with three expansion bases and (b) result with five expansion bases

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