Xingchen LiuSpatial Automation Laboratory,

University of Wisconsin-Madison,

Madison, WI 53706,

e-mail: xingchen@wisc.edu

Vadim Shapiro1

Spatial Automation Laboratory,

University of Wisconsin-Madison,

Madison, WI 53706

e-mail: vshapiro@wisc.edu

Sample-Based Synthesis ofFunctionally Graded MaterialStructuresSpatial variation of material structures is a principal mechanism for creating and con-trolling spatially varying material properties in nature and engineering. While the spa-tially varying homogenized properties can be represented by scalar and vector fields onthe macroscopic scale, explicit microscopic structures of constituent phases are requiredto facilitate the visualization, analysis, and manufacturing of functionally graded mate-rial (FGM). The challenge of FGM structure modeling lies in the integration of these twoscales. We propose to represent and control material properties of FGM at macroscaleusing the notion of material descriptors, which include common geometric, statistical,and topological measures, such as volume fraction, correlation functions, and Minkowskifunctionals. 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 aprocess of selecting neighborhoods from a reference FGM, based on target materialdescriptors fields. The effectiveness of the proposed method in generating a spatiallyvarying structure of FGM with target properties is demonstrated by two examples: designof a graded bone structure and generating functionally graded lattice structures with tar-get volume fraction fields. [DOI: 10.1115/1.4036552]

1 Modeling of Functionally Graded Material

Structures

1.1 Functionally Graded Materials. Functionally gradedmaterials (FGMs) are heterogeneous materials made of two ormore constituent phases with properties changing gradually withposition [1]. The spatial variation (gradation) of properties isdetermined by changes in the geometric configuration of theconstituent phases (also known as material structure) that serveparticular mechanical (or more generally, physical) function.FGMs are ubiquitous in nature due to their abilities to satisfymultifold functional constraints and adapt to conflicting require-ments; this also explains why FGMs are a popular candidate asengineering materials. FGMs may be found in a broad spectrumof applications, including filters [2], wear resistant coatings [3],bone and dental implants [4–6], and cutting tools [7,8]. Teeth[4], bones [5], and bamboos [9,10] are a few examples of FGMin nature.

Traditionally, the modeling of FGM focuses on the level ofmacroscopic properties [7,11,12]. However, it is the underlyingspatial variation of material structures that create and control thespatially varying properties. Recent advances in additive manu-facturing [1,13–15] and exponentially growing computing powerenables the shift of engineering design from the classical materialselection paradigm into a simultaneous design of material compo-sition and structure [16,17]. Systematic modeling and design ofgraded structures with desired mechanical properties is a keyingredient of this paradigm shift. In this paper, we study the syn-thesis of two-phase structures, such as porous structures, from areference material sample. The material structures are discretizedinto sets of voxels, which is consistent with the popular imageacquiring techniques such as microcomputed tomography (lCT)and magnetic resonance imaging (MRI).

1.2 Synthesis Problems in Material Structure Modeling.The basic construction problem of material structure modeling isillustrated in a commutative diagram as shown in Fig. 1. For thesample-based material structure synthesis, we are typically givena discretized reference material structure sample X and a targetmaterial properties field specified over some geometric domain.The effective material properties P(X) of the sample material Xcan be evaluated via map h: one of many known homogenizationmethods. For example, effective Young’s modulus can behomogenized through various homogenization theories, includingthe Gibson model for cellular materials [18], Voigt–Ruessbounds, Hashin–Strikman bounds [19], Green’s function-basedmethod [20,21], and power law [22], which is frequently used insolid isotropic material with penalization (SIMP)-based topologyoptimization [23]. Note that P(X) may be evaluated at any pointof p 2 R3 whether it belongs to X or not, because homogeniza-tion h typically requires integrating and averaging of themechanical properties in some neighborhood of p. We willdenote the property evaluated at a point p as P(p) when there isno ambiguity.

The target properties fields P(Y) can be designed by variousgradation techniques (map g), such as interpolation schemes in

Fig. 1 Key problem: construct a material structure Y with tar-get effective material properties distribution P(Y), given a mate-rial structure X with effective properties distributions P(X). Xand Y are usually represented by some microscopic models.Map h evaluates the effective properties of X and Y. Map g rep-resents gradation techniques to design the target propertiesfields on macroscopic scale. Map f symbolizes the integrationof macroscopic and microscopic representations.

1Corresponding author.Contributed by the Computers and Information Division of ASME for publication

in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscriptreceived August 25, 2016; final manuscript received April 12, 2017; publishedonline May 19, 2017. Assoc. Editor: Jitesh H. Panchal.

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solid modeling [7] and solid isotropic material with penalization(SIMP) [23] in topology optimization. Given the target macro-scopic properties distribution, the result structure Y can either becalculated from X by updating the parameters of the microscopicmodel or from P(Y) directly through inverse homogenization [24].It appears that the solutions to both construction problems requireknowledge of the inverse homogenization map h�1. Unfortu-nately, the relationship between the effective material propertiesand the corresponding material structures that produce them isknown only for some periodic and cellular structures but not ingeneral [25]. When such relationship is not known, an iterativeapproach is often adopted: the structure Y is repeatedly updatedand the properties h(Y) evaluated until they match the target P(Y).There are at least two problems with this approach. The homoge-nization procedure itself is a computationally expensive processthat may require additional material samples and is difficult toautomate. Furthermore, while the inverse homogenizationapproach completely disregards the input structure X, the micro-scopic model requires a complete understanding and parameter-ization of the generative process for the desired class of materialstructures, which is not readily available for many natural andsynthetic materials.

Recently, Liu and Shapiro [26] showed that Markov randomfield (MRF) texture synthesis techniques may be used to recon-struct stationary material structures. We recognize that MRF mod-els material structures as a collection of neighborhoods andtexture synthesis rearranges neighborhoods so that both the micro-scopic geometry and macroscopic statistics are preserved (themacroscopic statistics are constant fields in the case of stationarymaterial structures). In the Jordan curve example, we observe thatthe microscopic geometry is preserved even though the macro-scopic information is lost when the material structure is not sta-tionary. This inspires us to devise a mechanism to guide thesynthesis process by a target field, which controls macroscopicproperties distribution while preserving the microscopic geometryof the material structure.

1.3 Contributions and Outline. After reviewing the relatedmacroscopic properties models and microscopic material struc-ture models in Sec. 2, we propose the sample-based FGM struc-ture synthesis framework with its three pillars: construction byneighborhood, effective properties via descriptors, and texturesynthesis to rearrange neighborhoods in Sec. 3. A given materialsample X is modeled as a Markov random field (MRF), whichassumes the joint probabilities of material distribution are localand views X as a collection of neighborhoods (Sec. 3.1). To cir-cumvent the expensive numerical evaluation of neighborhoodeffective properties, we propose to represent and control materialproperties P(Y) of FGM in terms of material descriptors, a col-lection of geometric and topological measures that correlateclosely with many effective material properties (Sec. 3.2). Theproblem of synthesis of an FGM structure Y is formulated as aprocess of selecting neighborhoods in X according to the targetdescriptors fields. To solve the problem computationally, mate-rial structures are discretized on a regular lattice. Building onideas from texture synthesis, the FGM structure synthesis prob-lems are solved by combining the influence of material descrip-tor fields and adjoining neighborhoods (Sec. 3.3). When it ispossible, the proposed framework generates spatially varyingstructures with target material descriptors with given neighbor-hood size. But not all target properties P(Y) fields can beobtained using neighborhoods from a given sample X; hence, wediscuss the compatibility between the reference material X andtarget properties field P(Y), which is closely related to the qualityof synthesized structure Y (Sec. 3.4). The effectiveness of theproposed method is demonstrated by two examples: synthesis ofa graded bone structure and generating a functionally graded lat-tice structure with target volume fraction field (Sec. 4). Section 5concludes with a discussion on possible ways to speed up theimplementation.

2 Related Research

2.1 Macroscopic Material Composition Modeling. Model-ing of FGM properties has been a major research issue in solidmodeling. Such properties are usually represented by a collectionof scalar or vector fields defined over a geometric domain. With-out trying to be exhaustive, the following is a small sample ofwork in this area. Park et al. [11] use a volumetric multitexturingmethod representing a three-dimensional density gradient. Siu andTan [12] model both the distribution and the orientations of com-posite laminates through a source-based scheme. Biswas et al. [7]formulate the material function as an interpolation of propertiesassociated with given material features by inverse distanceweighting. Other representations of material function are largelyconsistent with different ways to represent solids, includingexplicit functions [27,28], implicit representations [29], and splinetensor products [28,30], to cite a few. Readers may refer Ref. [31]for a survey. In addition, the iterative updating of material compo-sition allows to optimize some material property functions forthermal stress [32], displacement [33], and structural weight [34].

Material composition modeling generally assumes the ability togenerate material structures from the designed material composi-tion [31]. However, a blend of the constituent phases by their vol-ume fractions cannot accurately represent the required spatiallyvarying properties, as it does not carry geometry or topologyinformation about the material structure [35–37]. Even thoughthese methods are unable to produce the material structure, theyare useful in modeling and design of the target descriptor andproperty fields used in this paper.

2.2 Material Structure Modeling. Existing approaches onmaterial structure modeling can be roughly divided into two cate-gories: procedural methods and reconstructive methods. In proce-dural modeling, material structure is produced by an algorithmicprocess. For example, Laguerre tessellations are used for ceramicand aluminum foams modeling [38]; FRep models both regularlattice and irregular porous material through optimizing in theparameters space of a pseudo-random model [39]; a texture map-ping system is designed to add internal structures into solids froma library [40]. Kou and Tan [41] use random Voronoi tessellation,with grains bounded by B-Splines to design irregular porous struc-ture with controllable pore shapes. Kou and Tan’s work isextended by grading the random distribution of Voronoi sites tomodel graded porous structure [42]. Procedural methods are effec-tive geometric modeling tools, but they suffer from two majorlimitations: (1) procedural methods require a complete under-standing and parameterization of the generative processes, whichare unknown for vast majority of natural and synthetic physicalprocesses and (2) direct relationship between procedural modelsand their effective bulk material properties is known only in a fewspecial cases, and cannot be established in general. These disad-vantages make procedural methods limit their applications in thedesign of FGM structures.

Reconstructive methods seek to generate material structuresdirectly from given descriptors and/or bulk material propertiesthat are estimated from given reference material samples. Themost common approach is to focus on matching material descrip-tors, especially two-point correlation functions, between a givenreference sample and the desired reconstruction result by solvinga large-scale stochastic optimization problem [43–45]. Oneadvantage of the optimization-based approach is its applicabilityto a broad range of random heterogeneous materials because itdoes not depend on the stochastic process by which a materialstructure is formed [36]. When the reference structure isabstracted by its descriptors, important structural (geometric andtopological) details of the reference material are usually lost. Thisimplies that optimization-based approaches will reconstruct simi-lar material structures for distinct reference materials with similardescriptors. Efforts to address this accuracy issue include usinghigher order correlation functions [46], including two-point

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cluster function into the target function of the optimization as ahybrid method [47], and implementing a dilation-erosion scheme[48]. However, once the geometric and topological informationabout the structure is lost, it cannot be fully recovered.

Another related problem is inverse homogenization, in which amaterial structure is constructed to satisfy prescribed positivesemidefinite constitutive tensors [24]. This problem appears to besolved only for a few periodic structures using unit cells. An FGMmay be constructed by designing each unit cell individually [49],which is computationally very expensive and impractical for realapplications. Designing unit cells with different effective constitu-tive properties is not the topic of this study.

Parametric [50–52] and nonparametric [26,53–55] texture syn-thesis techniques have recently been used for material reconstruc-tion by several researchers. In particular, Liu and Shapiro [26]demonstrated that nonparametric Markov random field texturesynthesis can be used to reconstruct a variety of periodic and ran-dom heterogeneous structures while preserving their geometric,topological, and physical properties. The classical texture synthe-sis can reconstruct only stationary material structures that remain(statistically) invariant under translation [56,57]. Such structurescannot be graded in a controlled fashion. Later advances in thefield [58–60] use an external control field to influence and to guideplacement of next pixel (patch or material element). This relaxesthe requirement of global stationarity in texture synthesis processand makes it controllable.

3 Problem Formulation

3.1 Construction by Neighborhoods. We model the refer-ence material structure X and the target structure Y as Markov ran-dom fields (MRFs). MRF assumes probabilities of materialdistribution are local, i.e., the material structure at any pointdepends on only the neighboring points. A refined version of thecommutative diagram in Fig. 1 is shown in Fig. 2. It acknowl-edges explicitly that the reference material structure X and the tar-get structure Y are collections of neighborhoods N(X) and N(Y),respectively. All neighborhoods are assumed to be of size s,selected to be large enough to give a meaningful description ofthe microscopic material structure.

We can now restate the material structure synthesis problem interms of neighborhoods: given a material structure X with effec-tive properties P(X), construct material structure Y such that (1)every neighborhood N(y) corresponds to some neighborhood N(x)in the sample material structure X, and (2) effective property ofsuch a neighborhood hðNðyÞÞ matches that specified by the targetproperty field P(y). The design objective can be cast into anoptimization problem: searching for Y such as the energy

E ¼ð

Y

½a minx2X

ðkNðxÞ � NðyÞkÞ þ ð1� aÞðkhðNðyÞÞ � PðyÞkÞ� dy

(1)

is minimized.The matching of the neighborhoods simply reflects the require-

ment that the geometry and composition of the material structureY originates from the reference structure X. The second conditionensures that the synthesized material structure satisfies the targetspatially varying effective material property field P(Y). Theweight a is selected by the user or an application, depending onthe relative importance of the two criteria.

In all cases, the material structure synthesis problem has beenreduced to the problem of matching and selecting neighborhoodsin the reference structure X and reassembling them in the targetstructure Y by a map t shown as a dashed arrow in Fig. 2. We alsoobserve that the reformulation in terms of neighborhoods reducedthe difficult inverse homogenization problem to a problem of fil-tering: only those neighborhoods in N(X) that have desired

material properties gðPðXÞÞ will be selected for use in the targetmaterial structure N(Y).

3.2 Effective Material Properties Via Descriptors. Anotherobstacle to solving the FGM structure synthesis problem is theneed to evaluate material property P on any neighborhood in thereference materials. This is a nontrivial task that usually involvessolving multiple boundary value problems [61], making the recon-struction procedure inefficient and impractical.

Instead, we propose to replace material properties P(X) by theirproxies: a collection of geometric, topological, and statisticalmaterial descriptors D(X) that are easily computable for a givenneighborhood N(x) of the material structure. Specific examples ofD(X) include correlation functions [36] and Minkowski function-als [62]. Numerous studies focus on relating the effective materialproperties with material descriptors because the latter can bemeasured directly. Examples include volume fraction andVoigt–Reuss bound [63], n-point correlation functions and elasticproperties [64], Minkowski functionals and osteoporosis [65], andtotal number of aggregates and performance of tires [66].

Using material descriptors allows us to transform the commuta-tive diagram in Fig. 2 into the one shown in Fig. 3. The map a rep-resents the relationship between material properties P(X) and theirmaterial descriptors D(X). Just like material properties, materialdescriptors D(X) are spatially varying fields that are computed onfixed-size neighborhoods in X and can be evaluated at any pointx 2 R3 yielding D(x). When map a is invertible, there is a one-to-one correspondence between properties P(X) and descriptorsD(X). With this assumption, selecting neighborhoods N(Y) pos-sessing properties P(Y) becomes equivalent to matching materialdescriptors D(Y). The latter task is much simpler because itinvolves only geometric computations. We note that numericalhomogenization techniques (for example, those based on the finiteelement method) can also be used to estimate material properties.This could be particularly useful in situations when relationshipbetween homogenized properties and other material descriptorshas not yet been established.

3.3 Neighborhood Rearrangement Via Texture Synthesis.To solve the proposed problem computationally, we discretize thematerial structures on an orthogonal lattice grid. The neighbor-hood of a point x 2 R3 is approximated by an orthogonal (rectan-gular in 2D) lattice neighborhood centered at x (Fig. 4)

N xð Þ ¼ x0 : kx0 � xk1 �s

2

� �(2)

where s is the given neighborhood size. With D(y) and map d asthe surrogates for P(y) and map h, respectively, the design objec-tive (Eq. (1)) becomes: searching for Y such as the energy

E ¼ð

Y

½a minx2X

ðkNðxÞ � NðyÞkÞ þ ð1� aÞðkdðNðyÞÞ � DðyÞkÞ� dy

(3)

is minimized, where minx2XðkNðxÞ � NðyÞkÞ finds the best match-ing neighborhood in N(X) and kdðNðyÞÞ � DðyÞk measures the

Fig. 2 Synthesis of graded material structures are reformu-lated in terms of neighborhoods

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difference between the actual descriptors of N(y) and targetdescriptors D(y).

The optimization problem (Eq. (3)) is NP–hard [67] and allexisting techniques compute approximations to the optimal solu-tions [56,68,69]. In this study, the graph-cut texture synthesisalgorithm [59,67] is adopted:

(1) Determine N(Y) neighborhood by neighborhood in steps ofs–o in raster (lexicographic) order (Fig. 4), where o is thesize of overlap between adjacent neighborhoods.

(2) For each neighborhood N(y), the best matching neighbor-hood in X is selected from N(X) to minimize the weightedcombination

dtotal ¼ adN þ ð1� aÞdD (4)

where dN measures the difference between N(y) and all neighbor-hoods in N(X) within the overlap region (dark shaded region inFig. 4)

dNðxÞ ¼X

ði;jÞ2overlap

jNiðxÞ � NjðyÞj (5)

and dD represents the weighted difference of all material descrip-tors between D(X) and D(y)

dDðxÞ ¼X

i¼1;2…k

wijDiðxÞ � DiðyÞj (6)

(3) Compute the voxelwise mismatch over the overlap region(Fig. 4). Use graph-cut algorithm [70] to find the optimalhypersurface (piecewise linear curves over the dark shadedregion in Fig. 4, a curve in 2D, and a surface in 3D) thatseparates the overlap region in two parts while minimizingthe accumulated mismatches along the boundary.

The algorithm solves Eq. (3) with the following three approxi-mations: (1) rather than for all neighborhoods in Y, the

neighborhoods and material descriptors constraints are checked insteps of s–o; (2) the total energy E is minimized in a “greedy”fashion: in raster order, the best fit is chosen for each neighbor-hood; (3) dðNðyÞÞ in Eq. (3) is approximated by dðNðxÞÞ, giventhe best matching N(x) eventually becomes N(y).

Even though the algorithm involves a significant amount ofapproximation, it is able to duplicate a stationary material struc-ture with guaranteed accuracy by consistently estimating the jointdistribution as well as material descriptors on a predefined neigh-borhood window [26,57]. The practical implication of this resultis twofold: (1) any neighborhood N(y) in the duplicated structureY almost surely exists in its counterpart reference structure X, i.e.,we expect that N(Y) � N(X); (2) the probability distributions ofsuch neighborhoods with regard to both structures are also thesame, as a direct consequence of the stationarity assumption. Incontrast, the quality of the constructed graded material dependson the compatibility between reference material and target fields,which is the subject of Sec. 3.4.

3.4 The Quality of the Synthesis Result. In a typical synthe-sis problem, target fields D(Y) may be created directly as a scalaror vector field via a map g that could be a combination of variousgradation techniques. It is not reasonable to expect that every suchproperty D(Y) can be realized by selecting neighborhoods in N(X).For example, when a target property does not exist in the refer-ence material or there are no adjacent neighborhoods in the refer-ence structure satisfying the properties of the adjacentneighborhood in the target field, the energy of Eq. (3) can neverdecrease to 0. If there exists Y satisfying both conditions in Eq.(3), i.e., energy E¼ 0, we say that the target fields D(Y) are com-patible with given reference material X and neighborhood size s.

The “binary” definition above is often too restrictive to be trueexcept for a few special cases, such as reconstructing a stationaryperiodic structure or when target descriptor fields D(Y) is exactlythe same as D(X) (see bone reconstruction example in Sec. 4.1).Nonetheless, it gives some theoretical backgrounds on what toexpect for the quality of the synthesized result Y: in real design sce-narios, a perfect solution usually does not exist due to the lack ofcompatibility between the inputs. To quantitatively assess the qual-ity of the result Y, we introduce the notion of (N, D)—compatible,where N measures the maximum mismatch between neighborhoods(dN) and D measures the maximum deviation from target descrip-tors (dD). It is noted that the same notion may extend to quantifythe degree of compatibility between X, s, and D(Y) by the qualityof the optimal solution of Eq. (3). Synthesis problems satisfyingthe original compatibility definition are (0,0)—compatible withthis new notion.

4 Experiment Results

4.1 Femur Bone Reconstruction. In this section, we demon-strate the effectiveness of the proposed approach for reconstruc-tion of a 2D (image of) femur bone structure (Fig. 5(a)).Minkowski functionals are the preferred material descriptors inthat they are standard measurements for bone structure characteri-zation [71]. There are exactly dþ 1 for a d-dimensional materialstructure: volume, surface area, average mean curvature, andEuler characteristic for d¼ 3, and area, perimeter, and Euler char-acteristics for d¼ 2. Regions of different properties in bone struc-tures can be differentiated by Minkowski functionals.

For ease of comparisons, both dN and dD in Eq. (4) are normal-ized into range ½0; 1�. Three Minkowski functionals are equallyweighted and a is chosen as 0.9. Both the reference and recon-struction results are discretized on a lattice of 399� 343 pixels,with a fixed size neighborhood of 25� 25 pixels. Figures5(b)–5(d) show the fields of Minkowski functionals D(X) for thereference bone structure X: area, perimeter, and Euler characteris-tic. Since this is a reconstruction problem, the target descriptorsfields D(Y) are identical to D(X). The inputs are (0,0)—compatible

Fig. 3 Problem formulation via material descriptors

Fig. 4 Synthesis of Y neighborhood by neighborhood in stepsof s–o. For this illustration, s 5 7 is the neighborhood size,o 5 2 is the size of overlap. Numbers in the center of the neigh-borhoods indicate the order of the synthesis. The piecewise lin-ear curve illustrates the hypersurface that separates theoverlap regions in two parts while minimizing the accumulatedmismatches along the boundary. Two piecewise linear curvesrepresent the hypersurfaces separating neighborhood 5 and 7from synthesized material structure, respectively.

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because X itself is the (0,0)—compatible solution we seek. Figures5(e) and 5(f) are reconstructions without and with the control ofMinkowski functionals fields, respectively. As expected, recon-structions with Minkowski functionals fields are noticeably betterthan reconstructions without it. We successfully recovered the Xsince we are only selecting the neighborhoods in Fig. 5(a) target-ing its own Minkowski functionals fields. It took 3.4 s for a rela-tively inefficient MATLAB implementation to reconstruct Fig. 5(f)on an Intel Core i7 3.4 GHz CPU.

4.2 Bone Structure Synthesis. Bone structures are vulnera-ble: diseases, physical injuries, and corticosteroids are among var-ious reasons leading to their degradation. In rare cases, thedeteriorated region needs to be removed and a bone scaffold maybe implanted to stimulate the bone to regenerate itself as well asto bear loads. An ideal design of scaffold should mimic the bonestructure in terms of mechanical properties, porosity, and shouldbe bio-compatible, bio-active, or bio-resorbable to enhance tissuegrowth [72,73]. With the assumption that a more natural shapewould be more stimulating to bone regeneration process, it is pos-sible to design the scaffold as the complement of the synthesizedbone structure.

Figure 6(a) shows a femur bone with deteriorated regionremoved. To design the target Minkowski functionals fields, weuse the inverse distance interpolation [7] of Minkowski function-als values on the existing bone structure. In other words, Minkow-ski properties at any point are represented as the weighted sum ofthe properties known at some locations (typically points or boun-daries), where the weights are inversely proportional to thepowers of distances to these locations

D yð Þ ¼XN

i¼1

D yið Þd y; yið Þp

�XN

i¼1

1

d y; yið Þp; 8i; d y; yið Þ 6¼ 0

D yið Þ; 9i; d y; yið Þ ¼ 0

8>><>>:

(7)

where d is the Euclidean distance between two points and each yi

represents the center of an existing neighborhood in Fig. 6(a).Figures 6(b)–6(d) show the interpolated target descriptors fieldswith the power of distance equals 4. For nonconvex domains,interpolation with interior distances may be advantageous [74].

Figures 6(e) and 6(f) are generated bone structures from the25� 25 neighborhoods available in Fig. 6(a): with the target Min-kowski functional fields calculated from the original intact bone(Fig. 5(c)) in 6(e), and with interpolated target Minkowski func-tional fields in 6(f). Note also that the proposed procedure connectsthe synthesized bone structures to the healthy part of Fig. 6(a)seamlessly. Mean and maximum differences between Minkowskifunctional values for the constructed results and those computed forthe original bone sample in Fig. 6(a) are summarized in Table 1.The results suggest that using inverse-distance interpolated fields astarget descriptor fields is a viable alternative to replacing the origi-nal descriptor fields (which are likely to be unknown). We expectthat the construction results can be improved further when addi-tional reference materials may be available for neighborhood selec-tion, for example, additional healthy bone samples from othersources.

4.3 Functionally Graded Lattice Structure. One obviousway to design the properties distributions of a functionally gradedmaterial is through optimization. We use solid isotropic materialwith penalization (SIMP), a popular method for topology optimi-zation, to design the distribution of stiffness inside the designdomain [75]. SIMP iteratively updates the volume fraction (rela-tive density) distribution inside the design domain to optimize thedesign objective, such as compliances, while meeting a set of con-straints, such as a constraint on the total volume of the part. Thepower law is used as the homogenization model to establish therelationship between material descriptor (volume fraction) andmaterial property (stiffness) while penalizing the intermediatedensities

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 Minkowskifunctionals as descriptors fields. Masks are used to stop the algorithm from scanning black regions surroundingthe 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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E ¼ Emin þ qpðEmax � EminÞ; q 2 ½0; 1� (8)

where Emax is the elastic modulus of the solid material, Emin is theelastic modulus of the void material, which is nonzero to avoidsingularity of stiffness matrix in finite element analysis, and p isthe penalty factor.

After the iteration converges, the continuous volume fractionfield is often thresholded to achieve a “0-1” design, but this is notnecessary for our application. Instead, with the proposed method,we can fill the design space with any given material structures—random or periodic—according to the volume fraction fielddesigned by SIMP. Here, we show that the proposed method per-forms equally well on periodic lattices.

Figure 7 demonstrates the synthesis of a graded material struc-ture that is functionally a cantilever beam with fixtures on the topand bottom left edges and downward loads on the bottom rightedge. The target volume fraction distribution (Fig. 7(a)) isdesigned by SIMP with an overall volume fraction of 0.5. Thelower and upper bounds of the volume fraction are 0.3 and 0.85.We set the penalty factor as one for a gradually varying volumefraction field. Figure 7(b) shows the reference material: a proce-durally generated spatially varying lattice. The synthesized

structure is shown in Fig. 7(c) with a neighborhood size s of 14�14� 14 and four being the size of overlap along every direction.In some applications, such as 3D printing, the connectivity of thestructure is very important: the generated structure is required tobe a single connected piece. We enforce the connectivity byrequiring every new neighborhood added to the synthesized struc-ture to be a single connected one and connected to the existingneighborhoods. Other desirable properties such as smallest featuresize can be guaranteed the same way by construction. It is notedthat not all volume fractions in the target fields are realized in thereference lattice. As a result, some of the target volume fractionsare achieved approximately. The compatibility of the result issummarized in Table 2. For comparison, Fig. 7(d) shows thegraded lattice generated using the same procedural method as thereference structure.

We observe that the reference lattice can be constructed by twotypes of unit cells (Figs. 8(a) and 8(b) that are dual to each other.The two types can be differentiated by their Euler characteristics v:v¼ 1 for type 1 and v ¼ �4 or 2 for type 2. The Euler characteristicof type 2 unit cell switches from �4 to 2 with increasing volumefractions. v ¼ �4 is calculated by gluing two triple torus (v ¼ �2)together over a two-dimension circle (v¼ 0): �4 ¼ �2� 2� 0.

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 boneregions, (e) and (f) show the synthesized bone structures different target descriptors fields. (a) Femur bone withthe deteriorated region removed, (b) interpolated perimeter field, (c) interpolated area field, (d) interpolated Eulercharacteristic field, (e) synthesis of missing bone structure with Minkowski functionals field from original bonesample (Fig. 5(c)), and (f) synthesis of missing bone structure with Minkowski functionals field interpolated fromexisting structures.

Table 1 Maximum and mean difference of Minkowski functionals between synthesized structures and original bone structuresover empty region. Minkowski functionals are rescaled to [0, 1].

Area Perimeter Euler characteristic

Max Mean Max Mean Max Mean

Figure 6(e) 0.2342 0.0178 0.1958 0.0121 0.3878 0.0467Figure 6(f) 0.3562 0.0236 0.2167 0.0177 0.4286 0.052

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With the Euler characteristics, we control the synthesis of the lat-tice structures by selecting neighborhoods of type 1 or type 2 unitcells only. The results are illustrated in Figs. 8(c) and 8(d),respectively.

Summarized in Table 2, the difference in neighborhoods is cal-culated as the ratio of the mismatched voxel to the volume of theoverlap region. The errors of Fig. 7(d) are smaller than Figs. 7(c)and 7(d). This is because that in Fig. 7(d), the Euler characteristicsare not constrained and there are more neighborhoods to choosefrom. Figure 9 shows an additional example of a wheel designwith fixtures on four lower corners and load on the center of thebottom surface. The connectivity of the neighborhood is notenforced for this example and small disconnected pieces can beseen near the edges. The neighborhoods with disconnected piecesare hardly mistakes as they do exist in the reference material. Thecomputation time of each example takes about 90 min, which issignificantly longer than the two-dimensional cases. This isbecause the algorithm is linear in time to not only the number ofneighborhoods to generate in Y, but also the number of neighbor-hoods to search for in X and the size of overlap and all of them aresignificantly larger in 3D. Possible acceleration schemes are dis-cussed in Sec. 5.

5 Conclusion

We formulated the problems of synthesizing functionallygraded material structures as a process of selecting neighborhoodsin a reference material sample. The new formulation allows toeffectively bypass the difficult problem of inverse homogenizationusing techniques from Markov random field texture synthesis. Wealso introduced the notion of material descriptors, and Minkowski

functionals in particular, to guide the construction of graded mate-rial structures, without the need to perform numeric homogeniza-tion or to solve boundary value problems. The fully implementedalgorithm supports synthesis of FGM structures by combining theinfluence of adjacent neighborhoods and target material descriptorfields in the neighborhood selection step of texture synthesis. Wedemonstrated the effectiveness of the proposed approach forreconstruction and synthesis of graded bone structures as well asgenerating functionally graded lattice structures.

Many acceleration schemes have been proposed for 2D synthe-sis, including PatchMatch [68], Image melding [76], and PatchT-able [69], whose core idea is to perform an approximate nearestneighborhood (ANN) search over N(X). However, ANN searchinevitably infects the quality of Y. In addition, most accelerationschemes work on two-dimension images and do not scale to threedimensions straightforwardly. To speed up without compromisingthe quality of the result, we may compute dN only for thoseneighborhoods with small dD. Obviously, searching for such asubset of N(X) will not jeopardize the quality of Y if D(Y) and X

Fig. 7 Synthesis of graded material structure that is functionally a cantilever beam with fixtures on the topand bottom left edges and downward loads on the bottom right edge. The target volume fraction field rangingfrom 0.3 to 0.85 is designed by SIMP with penalty factor equals 1. (a) Target volume fraction distribution from0.3 to 0.85, 30 3 16 3 8, (b) reference material, 80 3 80 3 80, (c) FGM structure synthesized by the proposedframework, 300 3 160 3 80, and (d) FGM structure generated by procedural method, 300 3 160 3 80.

Table 2 Errors and compatibility of the 3D synthesis results

Volume fraction Neighborhood

Max Mean Max Mean

Figure 7(d) 0.1004 0.0128 0.1450 0.0334Figure 8(b) 0.1700 0.0163 0.1253 0.0416Figure 8(d) 0.1378 0.0247 0.1327 0.0402Figure 9(d) 0.1481 0.0152 0.1791 0.0391

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Fig. 8 Functionally graded lattice structures for a cantilever beam with different unit cells: (a) Type 1 unitcell, (b) type 2 unit cell, (c) FGM structure with type 1 unit cell, and (d) FGM structure with type 2 unit cell

Fig. 9 Synthesis of graded material structure that is functionally a wheel structure with fixtures on four lower cor-ners and load on the center of bottom surface. The target volume fraction field ranging from 0.3 to 0.85 is designedby SIMP with penalty factor equals 1. (a) Volume fraction distribution, 20 3 20 3 10: view from top, (b) volume frac-tion distribution: view from bottom, (c) reference functionally graded lattice, 100 3 100 3 100, (d) FGM structuresynthesized by the proposed framework: top, Resolution: 200 3 200 3 100, and (e) FGM structure synthesized bythe proposed framework: bottom.

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are (0,0)—compatible. However, if not fully compatible, eliminat-ing too many candidate neighborhoods in N(X) could result inmore geometric discontinuities, which may in turn lead to undesir-able physical defects (e.g., stress concentrations) in the synthe-sized structure Y. This is the main reason a weighted combinationof dN and dD is adopted in Eq. (3). To conclude, filtering N(X) byD(Y) should depend on the degree of compatibility and requiresfurther studies.

An apparent limitation of the proposed method is that it cannotguarantee that the specified distribution of FGM properties isachieved, because this explicitly requires compatibility betweenthe reference material structure and the target properties field. Wenote, however, that the notion of (N, D)-compatibility allows finetuning the trade-offs between the neighborhood and the descriptorincompatibilities. Methods for evaluating the compatibilities a pri-ori appear to be a useful direction for future explorations. Rem-edies for large incompatibilities include modifying the targetproperties field, enriching or modifying the reference materialstructure, or both. For example, the reference material structuremay be modified using morphological operations to increase com-patibility, but such modifications could also alter the properties ofthe reference material. New reference material structures couldalso be designed by other methods reviewed in Sec. 2, when thetarget properties are beyond the reach of the direct modification ofthe reference material structure.

One open issue of the proposed approach is how to choosethe optimal neighborhood’s size to balance the accuracy of thesynthesis and computational cost. With the assumption of thegrain-germ model [77], when the sample material has constantintergrain distance, a constant size of neighborhoods may be cho-sen based on the two-point correlation function of the material, asproposed in Ref. [26]. For materials with varying of intergrain dis-tance, the optimal size of neighborhood should also be varying.Multiple iterations of the aforementioned algorithm may berequired to determine the optimal size of every neighborhood.

For material structures exhibiting anisotropy, both Minkowskifunctionals and volume fractions are rotation invariant and there-fore cannot capture the anisotropy of material structures. If theanisotropy of material is of interest, designers may opt for mate-rial descriptors that are sensitive to rotations, such as two-pointcorrelation function, or use the anisotropic material propertiesdirectly, as discussed in the end of Sec. 3.2.

Acknowledgment

This research was supported by NSF Grant Nos. CMMI-1344205 and CMMI-1361862 and the National Institute of Stand-ards and Technology. The responsibility for errors and omissionslies solely with the authors.

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