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Research Papers

Sample-Based Synthesis of Functionally Graded Material Structures

[+] Author and Article Information
Xingchen Liu

Spatial Automation Laboratory,
University of Wisconsin-Madison,
Madison, WI 53706,
e-mail: xingchen@wisc.edu

Vadim Shapiro

Spatial Automation Laboratory,
University of Wisconsin-Madison,
Madison, WI 53706
e-mail: vshapiro@wisc.edu

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 25, 2016; final manuscript received April 12, 2017; published online May 19, 2017. Assoc. Editor: Jitesh H. Panchal.

J. Comput. Inf. Sci. Eng 17(3), 031012 (May 19, 2017) (10 pages) Paper No: JCISE-16-2054; doi: 10.1115/1.4036552 History: Received August 25, 2016; Revised April 12, 2017

Spatial variation of material structures is a principal mechanism for creating and controlling spatially varying material properties in nature and engineering. While the spatially varying homogenized properties can be represented by scalar and vector fields on the macroscopic scale, explicit microscopic structures of constituent phases are required to facilitate the visualization, analysis, and manufacturing of functionally graded material (FGM). The challenge of FGM structure modeling lies in the integration of these two scales. We propose to represent and control material properties of FGM at macroscale using the notion of material descriptors, which include common geometric, statistical, and topological measures, such as volume fraction, correlation functions, and Minkowski functionals. At microscale, the material structures are modeled as Markov random fields (MRFs): we formulate the problem of design and (re)construction of FGM structure as a process of selecting neighborhoods from a reference FGM, based on target material descriptors fields. The effectiveness of the proposed method in generating a spatially varying structure of FGM with target properties is demonstrated by two examples: design of a graded bone structure and generating functionally graded lattice structures with target volume fraction fields.

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Fig. 1

Key problem: construct a material structure Y with target effective material properties distribution P(Y), given a material structure X with effective properties distributions P(X). X and Y are usually represented by some microscopic models. Map h evaluates the effective properties of X and Y. Map g represents gradation techniques to design the target properties fields on macroscopic scale. Map f symbolizes the integration of macroscopic and microscopic representations.

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Fig. 2

Synthesis of graded material structures are reformulated in terms of neighborhoods

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Fig. 3

Problem formulation via material descriptors

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Fig. 4

Synthesis of Y neighborhood by neighborhood in steps of s–o. For this illustration, s = 7 is the neighborhood size, o = 2 is the size of overlap. Numbers in the center of the neighborhoods indicate the order of the synthesis. The piecewise linear curve illustrates the hypersurface that separates the overlap regions in two parts while minimizing the accumulated mismatches along the boundary. Two piecewise linear curves represent the hypersurfaces separating neighborhood 5 and 7 from synthesized material structure, respectively.

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Fig. 5

Femur bone structure reconstruction (a) shows a cross section of a two-phase femur bone structure, (b)–(d) are Minkowski functionals fields of (a), (e) and (f) are reconstruction results without and with Minkowski functionals as descriptors fields. Masks are used to stop the algorithm from scanning black regions surrounding the bone. (a) Two-phase femur bone structure, (b) perimeter field, (c) area field, (d) Euler characteristic field, (e) reconstruction without Minkowski functionals fields, and (f) reconstruction with Minkowski functionals fields.

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Fig. 6

Synthesis of missing femur bone structures (a) shows the Femur bone with missing bone structures, (b)–(d) are target Minkowski functional fields designed by inverse distance interpolation from the healthy bone regions, (e) and (f) show the synthesized bone structures different target descriptors fields. (a) Femur bone with the deteriorated region removed, (b) interpolated perimeter field, (c) interpolated area field, (d) interpolated Euler characteristic field, (e) synthesis of missing bone structure with Minkowski functionals field from original bone sample (Fig. 5(c)), and (f) synthesis of missing bone structure with Minkowski functionals field interpolated from existing structures.

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Fig. 7

Synthesis of graded material structure that is functionally a cantilever beam with fixtures on the top and bottom left edges and downward loads on the bottom right edge. The target volume fraction field ranging from 0.3 to 0.85 is designed by SIMP with penalty factor equals 1. (a) Target volume fraction distribution from 0.3 to 0.85, 30 × 16 × 8, (b) reference material, 80 × 80 × 80, (c) FGM structure synthesized by the proposed framework, 300 × 160 × 80, and (d) FGM structure generated by procedural method, 300 × 160 × 80.

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Fig. 8

Functionally graded lattice structures for a cantilever beam with different unit cells: (a) Type 1 unit cell, (b) type 2 unit cell, (c) FGM structure with type 1 unit cell, and (d) FGM structure with type 2 unit cell

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Fig. 9

Synthesis of graded material structure that is functionally a wheel structure with fixtures on four lower corners and load on the center of bottom surface. The target volume fraction field ranging from 0.3 to 0.85 is designed by SIMP with penalty factor equals 1. (a) Volume fraction distribution, 20 × 20 × 10: view from top, (b) volume fraction distribution: view from bottom, (c) reference functionally graded lattice, 100 × 100 × 100, (d) FGM structure synthesized by the proposed framework: top, Resolution: 200 × 200 × 100, and (e) FGM structure synthesized by the proposed framework: bottom.

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