SANDIA REPORTSAND2019-10873Printed October 2019

Prepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185Livermore, California 94550

An Approach to Upscaling SPPARKSGenerated Synthetic Microstructures ofAdditively Manufactured MetalsJohn A. Mitchell

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ABSTRACTAdditive manufacturing (AM) of metal parts can save time, energy, and produce parts that cannototherwise be made with traditional machining methods. Near final part geometry is the goal forAM, but material microstructures are inherently different from those of wrought materials as theyarise from a complex temperature history associated with the additive process. It is well knownthat strength and other properties of interest in engineering design follow from microstructure andtemperature history. Because of complex microstructure morphologies and spatialheterogeneities, properties are heterogeneous and reflect underlying microstructure. This reportdescribes a method for distributing properties across a finite element mesh so that effects ofcomplex heterogeneous microstructures arising from additive manufacturing can besystematically incorporated into engineering scale calculations without the need for conducting anearly impossible and time consuming effort of meshing material details. Furthermore, themethod reflects the inherent variability in AM materials by making use of kinetic Monte Carlocalculations to model the AM process associated with a build.

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ACKNOWLEDGMENT

The work in this report was part of a sub-project focused on dynamic response of additivelymanufactured (AM) materials funded by the Advanced Certification and Qualification (ACQ)program. The author would like to gratefully acknowledge funding and support from KirkLevedahl, the ACQ program lead, Christopher Seagle (org 01646) the project manager, and PaulSpecht (org 01646) the project lead. The author would like to thank team members Dave Adams(org 01832), Jack Wise (org 01646) , Justin Brown (org 01646), Stewart Silling (org 01444) fortheir helpful discussions and support of this work. The author would also like to thank VeenaTikare for her support of this work.

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CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1. What are the challenges for AM materials in engineering design? . . . . . . . . . . . . . . . 71.2. Objectives for this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1. Synthetic microstructures for additive manufacturing process . . . . . . . . . . . . . . . . . . . 92.2. Computation of spatial statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1. Averaged two-point correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2. Fixed observation points and resolving spatial heterogeneity . . . . . . . . . . . . . 102.2.3. Clustering via an affinity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

LIST OF FIGURESFigure 1-1. Stainless steel micrographs. Top: wrought material with equiaxed microstruc-

ture. Bottom: AM material with heterogeneous microstructure. . . . . . . . . . . . . . . . 7Figure 2-1. SPPARKS model examples of AM process and associated microstructures; raster

patterns with and without serpentine; left column (with), right column (without). 13Figure 2-2. SPPARKS generated equiaxed microstructures in top row; bottow row shows

the average two-point correlation functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2-3. Evaluating spatial probability and average grain shape at an observation point.

Two microstructure realizations shown. The summation symbol ∑ abstractlyrepresents computation of average grain shape across many microstructure re-alizations by computing an average probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 2-4. Square plate with hole. FE mesh overlay with simulated microstructure. (Left):Simulated AM build microstructures on block/square of material; (Right): zoomon region of with FE mesh overlay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2-5. Computation of symmetric affinity matrix. Identify fixed observation pointswith FE quadrature points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 3-1. Underlying simulated microstructure with overlay of FE mesh sample. Nominalsample thickness 1mm used in dynamic response testing of AM material. . . . . . . 18

Figure 3-2. Rendering of upscaled synthetic microstructure for sample shown in Figure 3-1. 19Figure 3-3. Rendering of upscaled synthetic microstructure for sample shown in Figure 2-4

(Right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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1. INTRODUCTION

Additive manufacturing (AM) of metal components/parts shows great promise to decrease timeand fabrication costs; it also enables designs which cannot be realized using traditionalmachining. In practive however, AM introduces challenges that must be addressed before AMfabricated components/parts can be introduced as key elements in engineering designs.

1.1. What are the challenges for AM materials in engineeringdesign?

• AM materials exhibit spatial heterogeneity. Wrought materials typically consist ofequiaxed microstructures and exhibit elastic isotropy at relevant engineering length scales.In contrast, AM materials exhibit spatial heterogeneity and heterogeneous textures andmorphologies at various length scales. Heterogeneous residual stresses are inherent in AMmaterials due to complex thermal history.

• Measurements on AM materials indicate higher variability than traditionally wroughtmaterials in such things as yield stress, ductility, and ultimate stress.

• Challenges to modeling and design. Modeling and engineering design strategies must bedeveloped which integrate spatial heterogeneity and length scale effects inherent to AMmaterials.

Spatial heterogeneity and length scale differences between a wrought material and an AMmaterial are depicted in Figure 1-1. Microstructural morphologies, spatial heterogeneities, andlength scale differences between the wrought and AM materials are apparent.

Figure 1-1 Stainless steel micrographs. Top: wrought materialwith equiaxed microstructure. Bottom: AM material with het-erogeneous microstructure.

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1.2. Objectives for this work

The remainder of this report describes an approach for upscaling microstructure morphologies andspatial heterogeneities. The conceptual objectives of the upscaling strategy are listed below.

• Avoid vexing problem of meshing material details.

• Systematically represent properties on coarse continuum model.

• Respect microstructural morphologies and associated length scales.

• Reflect microstructure heterogeneity and AM material variability.

• Enable microstructure aware simulations for dynamic response of AM metallic materials.

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2. APPROACH

The approach used to systematically represent properties on a coarse continuum model consists ofthe following key elements. Details of these elements will be discussed in subsequentsubsections.

• SPPARKS generated synthetic microstructures based on model of AM process

• Computation of spatial statistics associated with observation points

• Application of cluster algorithms from machine learning to represent properties on coarsecontinuum models

Each of the above items are discussed in the following subsections.

2.1. Synthetic microstructures for additive manufacturingprocess

Kinetic Monte Carlo (KMC) simulations with SPPARKS [1, 2] are used to simulate the additivemanufacturing process. SPPARKS has been sucessfully used to simulate welding and additivemanufacturing processes [3, 4] and produces representative microstructures.

SPPARKS generated synthetic microstructures predict grain size and shape morphologiesproduced by a particular manufacturing process; microstructure morphologies are processsensitive. KMC models used to simulate process approximately represent a spatial temperatureprofile by estimating the size and shape of molten metal pools which can be observed in thelaboratory; pool size and shape strongly depend upon AM laser power and raster speed which areprocess specific. It is known that microstructures and measured engineering properties vary frombuild to build even for a fixed process; the exact cause of variability is not known although graingrowth and evolution is a stochastic process. Synthetic microstructures generated by SPPARKSintroduce build to build variability using a random seed so that each synthetic microstructure is arepresentative of the random process associated with the AM build.

A simple demonstration of process effects on microstructures is depicted in Figure 2-1; twosomewhat similar processes are depicted; all things equal except for slightly different rasterpatterns. In the left column, a horizontal raster pattern with serpentine is depicted; in the rightcolumn a raster pattern without serpentine is shown. Note similar microstructure morphologieswith the exception of center line where microstructures on the left are conceptually mirrored withrespect to the right. Hundreds of simulations are used and conceptually assembled to computespatial statistics.

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2.2. Computation of spatial statistics

Computation of spatial statistics are a key aspect of the method described in this paper. In thissection two-point correlations and a similar concept for fixed observation points will beintroduced and discussed. Fixed observation points logically extend two-point correlations andbetter handle spatial heterogeneity.

2.2.1. Averaged two-point correlations

In an earlier work, Castillo [5] used SPPARKS to generate synthetic microstructures andsubsquently computed two-point stastistics averaged across an entire microstructure; subsquentlyprincipal component analysis (PCA) was used to link process parameters to to principal values. Agood example of the averaged two-point correlation function is depicted Figure 2-2 for twomicrostructures. The two-point correlations can be quickly computed using fast Fouriertransforms (FFT). The following observations characterize two-point correlations for equiaxedmicrostructures.

• For any given point p within the microstructure, the averaged two-point correlation isconditional probability that surrounding points q are part of the same grain at p

• On an equiaxed microstructure, the two-point correlation is a radial function; at |r|= 0probability is 1; at large |r|, probability is 0

• On an equiaxed microstructure, the two-point correlation visually captures grain size

If strength analyses are conducted on AM materials wherein the length scale of grains is roughlythe same as that of the analyses, then a short coming of the above method is that it averages awayspatial heterogeneity which may be present. In AM, laser raster scans produce patterns inmicrostructure morphologies. Using the averaging approach to compute two-point correlations,these patterns are largely lost. Because of spatial heterogeneity, grains in one part of the build,e.g. along an axis A, may have different shapes and sizes in another area of the build, e.g.different relative position to axis A. When grains from different areas are aggregrated to form anaverage two-point correlation, some important information is lost.

2.2.2. Fixed observation points and resolving spatial heterogeneity

As noted above, spatial heterogeneity is largely lost when using the FFT approach to computetwo-point correlations. This is because the two-point correlation is computed as an average grainshape across all grains in an image or set of images without consideration of grain location. Analternative approach is to compute an average grain shape for a set of fixed observation pointswithin images. To generate an average shape at a particular observation point, manymicrostructure realizations must be synthesized. Instead of averaging grain shape across a singleimage, grain shapes are averaged across multiple microstructure realizations for a set of fixedobservation points.

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Let D⊂ R3 be a bounded spatial domain representing the volume over which a microstructure isknown; the volume is discretized with voxels. A microstructure is a collection of grains G whichfill the volume D. Each grain within the microstructure G is a collection of voxels which definethe spatial extent of the grain; each voxel is a member of one grain only. An observation point is aparticular voxel. Given two observation points, r,s ∈G, a grain membership function g(r,s) isdefined as

g(r,s) =

{1, if r,s part of same grain0, otherwise.

(1)

Using the membership function in (1), the average grain shape at any observation point r ∈G canbe computed. Consider a set S of N microstructure realizations, each with its own uniquecollection of grains GI . Associated with each microstructure realization, there is a membershipfunction gI as defined in (1). Each realization GI ∈ S,∀I ∈ [1,N] occupies the volume D ofinterest. See upper left microstructure graphic in Figure 2-3 for an example of a particularmicrostructure realization GI , where r,s ∈ D denote two observation points. The relative positionof s with respect to r is ξ = r− s. In this particular microstructure GI , the observation points r ands are located within different grains, denoted by orange and green respectively, hence g(r,s) = 0.If this was the only microstructure realization, then the probability that observation point s is partof the grain at r is p(ξ ) = 0, since they are in different grains. If there are N microstructurerealizations, an arithmetic average probability can be evaluated by counting the number of times nthat r and s are part of the same grain; in this case the probability p(ξ ) = n/N. If this concept isextended to all potential observation points relative to r, then the most likely grain shapeassociated r can be computed. This process is conceptually illustrated in Figure 2-3.

2.2.3. Clustering via an affinity matrix

Continuum scale finite element (FE) calculations are the intended application of the two-pointcorrelations described in previous subsection. With the two-point correlations, a spatial affinitymatrix is constructed and used to create synthetic microstructures by spectral clustering.

Spectral clustering used here relies on the property that for fixed observation points r,s, seeFigure 2-3, the probability that point r is part of the same grain at s, is the same as the probabilitythat point s is part of the same grain at r. This property expresses a symmetry which produces asymmetric affinity matrix and is expressed mathematically as p[r]〈ξ 〉= p[s]〈−ξ 〉.

Calculation of the affinity matrix requires overlay/registration of the FE mesh with a simulatedmicrostructure; an example is shown Figure 2-4. In this example, microstructure is simulated foran AM build of a block of material; a hole is subsequently machined from the block of material.It is desirable to run an FE simulation on the AM material with a hole. By imprinting andoverlaying the FE mesh onto the simulated material, a synthetic microstructure can be createdwhich represents both the build process and geometry of final test specimen.

Gauss quadrature points associated with each finite element are taken as fixed observation points;let n denote the number of observation points. Then the affinity matrix is a square matrix of sizen; row i of the matrix represents the probability that the remaining n−1 observation points arepart of the grain at observation point i. The matrix is symmetric; it may even be banded

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depending on grain sizes and shapes and how points are ordered. The affinity matrix and itsrelation with FE mesh is schematically depicted in Figure 2-5.

On a practical note, simulated microstructures are generally represented on a much finerresolution length scale than the FE mesh resolution, i.e. distance between observation points ismuch higher than distance between lattice points used to simulate microstructures in SPPARKs.However, observation points do not generally coincide with the lattice sites. Using spatial search,each observation point is associated with a lattice site which is then mapped to a grain id.

Probabilites at observation points were computed per Section 2.2.2 using approximately N = 150microstructure simulations; an example of one such simulation is shown on (Left) in Figure 2-4.These simulations were used to construct the affinity matrix; spectral clustering was calculatedusing scikit-learn [6]. To run the clustering calculation, the number of expected clusters nc mustbe specified. This provides the clustering algorithm a way to partition observation points intoclusters according to probabilties supplied by affinity matrix. In each microstructure simulation k,the number of unique grain ids associated with the set of observations points is denoted by nk

c.The number of clusters nc was calculated as the average of the set S = nk

c, k = 1,2, . . . ,N.Interestingly, S is a dataset for which statistics can be calculated, since for each microstructuresimulation k, a different number of clusters nk

c is potentially found. In this report, the averagevalue across the entire number of runs was used for spectral clustering calculations. If a differentvalue for nc, say for example the mean value plus one standard deviation was used, then slightlydifferent spectral clustering results would be obtained. This is an area of further investigation thatwill be reported on later.

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Figure 2-1 SPPARKS model examples of AM process and asso-ciated microstructures; raster patterns with and without serpen-tine; left column (with), right column (without).

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Figure 2-3 Evaluating spatial probability and average grainshape at an observation point. Two microstructure realizationsshown. The summation symbol ∑ abstractly represents compu-tation of average grain shape across many microstructure real-izations by computing an average probability.

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Figure 2-4 Square plate with hole. FE mesh overlay with sim-ulated microstructure. (Left): Simulated AM build microstruc-tures on block/square of material; (Right): zoom on region ofwith FE mesh overlay.

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Figure 2-5 Computation of symmetric affinity matrix. Identifyfixed observation points with FE quadrature points.

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3. EXAMPLES

Using the approach described in Section 2, two synthetic microstructures were created. Theseexamples are representive of the size and shape of material samples dynamically tested in thelaboratory on this project. The synthetic microstructures so generated were exported to an exodusfile. A new ALEGRA [7] capability, details of which are not described in this report, has beencreated for reading the synthetic microstructures for use in simulations of the material underdynamic loading conditions. This capability facilitates distributing properties such as moduli andyield strength across a mesh that reflects grain morphologies and length scales. Reporting onthese simulations will be done elsewhere.

The first example is a 2D cross-section of a circular disk with thickness 1.5mm, see Figure 3-1.Such a material sample may be tested in a gas-gun type experiment [8] or in electromagneticcompression [9]. Figure 3-1 contains an FE mesh of a test sample overlaid with onemicrostructure simulation of an AM build. As described earlier, many such microstructuresimulations would be used to create the most likely representative microstructure. The resultingsynthetic microstructure represented on mesh is shown in Figure 3-2.

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Figure 3-1 Underlying simulated microstructure with overlay ofFE mesh sample. Nominal sample thickness 1mm used in dy-namic response testing of AM material.

The second example, shown in Figure 2-4 (Right), is a square block of AM material with acircular hole cut out after manufacture. Dynamic testing of this sample is planned. Thedemonsrtation synthetic microstructure is shown in Figure 3-3. This example demonstrates thenovelty of the upscaling method to automatically capture geometry of a part – in this case theability to reflect the most likely microstructures around a hole geometrically captured by the FEmesh.

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Figure 3-2 Rendering of upscaled synthetic microstructure forsample shown in Figure 3-1.

Figure 3-3 Rendering of upscaled synthetic microstructure forsample shown in Figure 2-4 (Right).

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4. CONCLUSIONS

A new approach to upscaling microstructure morphologies associated with AM materials waspresented; the upscaled representations are intended to enable microstructure aware simulations.A new capability has been developed in ALEGRA which enables use of the upscaledrepresentations for the purpose of running microstructure aware simulations. These simulationsallow for distributing properties across a finite element mesh that respects AM materialmorphologies and length scales. An upscaled representation was demonstrated on a 2D plate witha circular hole. Using these upscaled representations of microstructures in computational modelswill help to understand microstructure effects on dynamic response of AM materials.

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REFERENCES

[1] Steve Plimpton, Aidan Thompson, and Alex Slepoy. SPPARKS, 2016.http://spparks.sandia.gov.

[2] Steve Plimpton, Corbet Battaile, Mike Chandross, Liz Holm, Veena Tikare, Greg Wagner,Ed Webb, and Xiaowang Zhou. Crossing the mesoscale no-man’s land via parallel kineticmonte carlo. Technical Report SAND2009-6226, Sandia National Laboratories, October2009.

[3] Theron M Rodgers, Jonathan D Madison, and Veena Tikare. Simulation of metal additivemanufacturing microstructures using kinetic Monte Carlo. Computational Materials Science,135:78–89, July 2017.

[4] Theron M Rodgers, John A Mitchell, and Veena Tikare. A Monte Carlo model for 3D grainevolution during welding. Modelling and Simulation in Materials Science and Engineering,25(6):1–22, 2017.

[5] Andrew Castillo, John A Mitchell, and Stephen Bond. Using and developing novel dataanalytics for digitalized microstructures. Technical Report SAND2016-12140, SandiaNational Laboratories, November 2016.

[6] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel,P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher,M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of MachineLearning Research, 12:2825–2830, 2011.

[7] A. C. Robinson et al. ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial,multiphysics code. Reno, Nevada, January 2008. Proceedings of the 46th AIAA AerospaceSciences Meeting. AIAA-2008-1235.

[8] Jack L Wise, David P Adams, Erik E Nishida, Bo Song, Michael Christopher Maguire, JayCarroll, Benjamin Reedlunn, Joseph E Bishop, and TA Palmer. Comparative shock responseof additively manufactured versus conventionally wrought 304l stainless steel. In AIPConference Proceedings, volume 1793, page 100015. AIP Publishing, 2017.

[9] E. M. Waisman, D. B. Reisman, B. S. Stoltzfus, W. A. Stygar, M. E. Cuneo, T. A. Haill, J.-P.Davis, J. L. Brown, C. T. Seagle, and R. B. Spielman. Optimization of current waveformtailoring for magnetically driven isentropic compression experiments. Review of ScientificInstruments, 87(6):063906, 2016.

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