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International Journal of Cast Metals Research

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Determination of pattern allowances forsteel castings using the finite element inversedeformation analysis

Daniel Galles, Jia Lu & Christoph Beckermann

To cite this article: Daniel Galles, Jia Lu & Christoph Beckermann (2019) Determination of patternallowances for steel castings using the finite element inverse deformation analysis, InternationalJournal of Cast Metals Research, 32:3, 123-134, DOI: 10.1080/13640461.2018.1558562

To link to this article: https://doi.org/10.1080/13640461.2018.1558562

Published online: 17 Dec 2018.

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Determination of pattern allowances for steel castings using the finiteelement inverse deformation analysisDaniel Gallesa, Jia Lub and Christoph Beckermannb

aOak Ridge Associated Universities, Oak Ridge, TN, USA; bDepartment of Mechanical Engineering, University of Iowa, Iowa City, IA, USA

ABSTRACTIn recent years, the development of computational models to predict casting distortionshas led to improvements in efficiency and accuracy over traditional pattern design thatrelies on shrink rules. Unfortunately, the determination of pattern dimensions usingcomputer simulation remains a trial-and-error process that requires several design itera-tions. In this study, the finite element inverse deformation analysis is utilized to calculatethe pattern geometry in a single iteration for a plastically deformed (i.e., distorted) body.This method is evaluated through a loop test, for which an inverse simulation is per-formed first to calculate the pattern shape. This configuration is then used as the inputgeometry for a forward simulation, which is shown to successfully recover the original as-cast shape used for the inverse analysis. Through this sequence, the inverse deformationmethod is shown to be a viable technique for the determination of pattern allowances inproduction castings.

ARTICLE HISTORYReceived 10 September 2018Accepted 6 December 2018

KEYWORDSInverse deformationanalysis; finite element;pattern allowances; casting;distortions; steel

Introduction

Casting distortions are unintended deformations thatoccur during metalcasting. Their presence may con-siderably impact pattern allowances (PA):

PA½%� ¼ feature lengthpattern � feature length as�cast

feature lengthpattern� 100

(1)

Eq. 1 defines a common metric used by industry toquantify dimensional differences between patternand as-cast features. For distortion-free castings, pat-tern allowances adhere to well-established shrinkrules based on the casting’s known thermal expansionbehaviour. For this case, pattern allowances areknown a priori and determination of the patternshape is trivial. Unfortunately, casting processes cangenerally be expected to produce distortions, whichin turn may lead to dimensional inaccuracies in the as-cast part.

Casting distortions are attributed to mechanicalstresses, thermal stresses, or some combination ofboth. Mechanical stresses arise from contact interactionsbetween the casting and mould (e.g. a sand corerestrains thermal contractions in the casting), whereasthermal stresses are the result of uneven cooling in thecasting and lead to spatially varying thermal strain rates.For both cases, permanent (i.e. plastic) deformationsoccur when stress levels exceed the yield strength ofthe casting material. As a result, feature dimensions maynot adhere to customer specifications. For this reason,pattern design is inherently an iterative process for

which the patternmaker alters the pattern througha trial-and-error process until the desired dimensionsare achieved. For each iteration, a casting is producedand then evaluated through a dimensional analysis.Such a design strategy is not only expensive (due tothe high energy costs associated with casting) but alsotime-consuming. Furthermore, the efficiency of the pro-cess heavily relies on the patternmaker’s expertise,which can vary considerably.

In recent years, the remarkable increase in computerspeeds has coincided with advancements in computa-tional codes suitable for predicting stresses and distor-tions during casting. In particular, researchers havedeveloped constitutive relations [1–13] for mechanicalmodels that account for complex physical phenomena(e.g. hot tears, macrosegregation, rate and hardeningeffects) during and after solidification for steels [1–6],aluminium alloys [7–12], and magnesium alloys [13].Additionally, complemental experimental studies [14–25] have provided awealth of data needed to determineparameters for these constitutive relations. Several stu-dies [14–17] relied on mechanical testing in laboratorysettings. However, because the microstructures ofreheated specimens may considerably differ fromthose created during solidification and cooling, othersused in situ measurements to study distortions [18–25].Recently, Galles and Beckermann [26–28] used in situmeasurements to calibrate constitutive model para-meters over a range of temperatures and loading con-ditions for ASTM 216 WCB steel [26] and bonded silicasand [27,28], which is commonly used to construct

CONTACT Daniel Galles daniel.j.galles.ctr@mail.mil Oak Ridge Associated Universities, Oak Ridge, TN 37830, USA

INTERNATIONAL JOURNAL OF CAST METALS RESEARCH2019, VOL. 32, NO. 3, 123–134https://doi.org/10.1080/13640461.2018.1558562

© 2018 Informa UK Limited, trading as Taylor & Francis Group

moulds. The predictive capability of this model was laterdemonstrated through a case study [29] in which dis-tortions were accurately predicted for a productioncasting.

The aforementioned studies have paved the wayfor a paradigm shift from physically to digitally-baseddesign within industry. Today’s casting simulationscan be performed with good accuracy and ina fraction of the time needed to produce and analysea physical casting. As a result, considerable time andcost savings are realized. Moving forward, additionalimprovements in accuracy and efficiency can beexpected to further increase the role computer simu-lation for pattern design.

Despite the benefits, digitally-based pattern designis similar to its physical counterpart in that both areiterative procedures. This is due to the fact that thekinematics of finite element codes are typically imple-mented in a forward framework. In other words, thefinite element simulations start from a reference con-figuration (i.e. pattern shape) and, based on the load-ing history, calculate the deformed configuration (i.e.as-cast shape). Hence, pattern design, whether physi-cally or digitally-based, remains a trial-and-errorapproach whose efficiency is predicated on the exper-tise of the patternmaker/design engineer.

Pattern design belongs to a class of engineeringapplications in which the unknown reference geo-metry must be determined. To this end, Yamada[30] and Govindjee et al. [31] pioneered an inversemethod that directly solves the equilibrium bound-ary condition for an elastic material. These seminalstudies spurred the development of inverse finiteelement implementations that were capable ofdetermining the reference configuration firstly forelastic deformations and later for elastoplasticdeformations. A comprehensive review of the for-mulation for elastic deformation is given by Lu andLi [32]. Unfortunately, a caveat for the elastoplasticproblem is that the deformations are history depen-dent and thus, the inverse deformation problem isgenerally ill-posed. A unique solution is onlyachieved when the loading history or the plasticstrain in the deformed state is known, as demon-strated by Germain et al. [33]. The authors alsoproposed a recursive procedure wherein the defor-mation and plastic variables were determined sepa-rately through nested iterations of inverse andforward analyses [34]. Lu and Li [35] further demon-strated that an inverse boundary value problemgives reasonably accurate results for both displace-ment and plastic variables, provided the deforma-tions are moderately large and the loading history isnearly monotonic. In general, casting processes canbe expected to fall under these conditions, as 1)loading is predominantly driven by (nearly) mono-tonic cooling and 2) the associated thermal strains

induce dimensional changes typically on the orderof a few percent. An exception occurs during solidstate phase transformations, when transformation-induced volumetric expansion may lead to unload-ing. However, the inverse solution will be applicableas long as minimal plastic deformations occur dur-ing the transformation.

In this study, the direct approach of inverse ana-lysis developed by Lu and Li [35] is utilized for plas-tically deformed (i.e. distorted) material bodies inorder to determine pattern dimensions for a steelcasting. A simple rectangular bar is analysed first todemonstrate the feasibility of the technique. Then,elastoplastic deformations are calculated fora realistic casting geometry consisting of a steelplate outfitted with risers and a gating system. Thedesired as-cast shapes are shown Figure 1 and serveas inputs for the inverse analysis. The mould isexcluded from the stress analyses in order to preventnumerical difficulties encountered with contact inter-actions between the casting and mould.Consequently, deformations in this study areinduced exclusively by thermal stresses. For thesimulations, a sequential thermal-mechanical analy-sis is performed in which the transient temperaturefields are calculated first using casting simulationsoftware and then used as inputs for the forwardand inverse stress analyses. Loop tests are conductedfor the analyses, in which the inverse problem issolved first to determine the pattern shape, whichis subsequently used as the input geometry for theforward analysis. The inverse technique is evaluatedby the forward problem’s ability to recover the ori-ginal geometry used for the inverse analysis. Finally,the accuracy of the inverse elastoplastic solution isevaluated through an error analysis.

Inverse elastoplastic problem and properties

The inverse elastoplastic problem presented in thissection calculates the pattern geometry from theknown as-cast shape. The model reviewed here isthe general quasi-static formulation presented by Luand Li [35] with the following simplifications. Due tothe relatively small casting sizes, body forces will havea negligible impact on distortions and therefore, arenot considered. Also, the exclusion of the mould fromthe mechanical problem precludes the need toinclude surface tractions in the formulation. Thesesimplifications reduce the boundary value problem to

σij;j ¼ 0 in Ω

Φ ¼ �Φ on @Ωu(2)

where σij is the Cauchy stress and �Φ is a prescribeddisplacement on boundary ∂Ωu. Equation 2 deter-mines an inverse motion Φ : Ω ! B 2 R3 for the

124 D. GALLES ET AL.

sought reference configuration, B, when starting fromthe given current configuration, Ω. Stated anotherway, the inverse deformation Φ (x, t) describes thecorrespondence between the current coordinatex and its reference coordinate X. Such a relation isthe kinematic inverse of the usual forward deforma-tion φ (X, t) of a material point that progresses fromthe reference coordinate to the current coordinate.Similarly, the inverse deformation gradient f = ∂x Φ(x, t) is the kinematic inverse of the forward deforma-tion gradient F= ∂Xφ (X, t). Additional details, includ-ing the finite element implementation, can be foundelsewhere [35].

During the inverse solution procedure, the materialresponse obeys a Hencky constitutive relation, whichis very close to a linear elastic response at smallstrains, provided the stress levels do not exceed theyield strength of the casting. For the elastic proper-ties, a constant value of 0.3 was specified for Poisson’sratio and a temperature-dependent Young’s modulus,shown in Figure 2(a), was taken from Koric andThomas [36]. For the case of yielding, the followingelasto-plastic constitutive relation is invoked:

σ ¼ σ0 1þ εp� �n

(3)

where σ is the von Mises stress, σ0 is the initial yieldstress, εp is the equivalent plastic strain, and n is thehardening exponent. Equation 3 is a simplified form ofthe elasto-visco-plastic constitutive relation used byGalles and Beckermann [20] to calculate stresses anddistortions during steel casting. The simplificationinvolves using a representative constant strain rate to

replace the rate dependent term in the original elasto-visco-plastic relation. The remaining temperature-dependent model parameters (σ0 and n) in Equation 3were determined using a Levenberg-Marquardt non-linear least squares algorithm and are shown in Figure2(b). A sufficient number of data points for this algo-rithm were generated at various temperatures from thecalibrated model of Galles and Beckermann [20] usinga representative strain rate of 1 × 10−5 s−1. Since theconstitutive parameters were based on a previously cali-brated casting model, the simulations in this study canbe expected to give reasonable predictions. Thermalstrains were calculated using the temperature-dependent linear thermal expansion coefficient shownin Figure 2(c). This curvewas calibrated from the thermalcontraction data of an unrestrained steel bar [20].

Numerical examples

Introduction

Casting processes occur over large temperature ranges,which induce thermal strains that may lead to consider-able stresses and associated distortions. As previouslystated, the present study only considers thermal stressesthat are created by uneven cooling in the casting and notmechanical stresses caused by contact interactionsbetween the casting and mould. For simple casting geo-metries without internal features (e.g. holes), minimalmechanically-induced distortions can be expected aftersolidification, as the casting quickly gains the necessarystrength to prevent any distortions created by mould-metal interactions [22]. This rationale motivated the

Figure 1. Geometries for the bar (a) and plate (b) castings. Dimensions in mm.

INTERNATIONAL JOURNAL OF CAST METALS RESEARCH 125

designs of the bar and plate casting geometries shown inFigure 1(a,b), respectively. Dimensions in the figures areshown inmm. For the bar, preliminary heat transfer simu-lations revealed nearly constant cooling. Therefore, a chillwas added to the bottom of the bar in order to generatedirectional cooling and induce ample thermal stresses.Such a strategy was not needed for the plate castingsystem, whose inclusion of simplified risers and a gatingsystem produced the varying section thicknesses neces-sary to naturally create uneven cooling.

The numerical examples presented in this sectionwere conducted utilizing a sequential thermal-mechanical coupling. Heat transfer simulations wereperformed first to calculate the transient temperaturefields, which were then used as inputs for the forwardand inverse stress analyses. A loop test served to eval-uate the inverse method. For this test, the inverseanalysis was performed first to determine the patternshape, which was subsequently used as the input geo-metry for the forward analysis. Validation of the inversetechnique is based on the loop test’s ability to recoverthe initial geometry used for the inverse analysis.

Heat transfer simulations

Spatial temperature gradients (i.e. uneven cooling)within the castings are essential for this studybecause they drive the thermal loading that

generates distortions. Without them, deformationscaused only by stress-free thermal contractions, inwhich case all dimensions will adhere to the pat-ternmaker’s shrink rules. Temperature fields werecalculated using the casting simulation softwareMAGMASOFT® [37]. In order to generate realisticcasting temperatures, the bonded sand mould wasincluded in the simulation model. Simulation inputsincluded thermophysical properties for the mouldand casting, solid volume fraction during solidifica-tion, and latent heat of solidification. In addition,the interfacial heat transfer coefficient must be spe-cified. This parameter accounts for the air gap for-mation between the casting and mould and thus,allows for decoupling of the thermal-mechanicalproblem. All inputs, excluding the latent heat ofsolidification, are temperature-dependent. The cali-bration procedure for the current heat transfermodel, as well as the aforementioned inputs, canbe found in Galles and Beckermann [38]. Due shortpour times, mould filling had a negligible impact onthe calculated casting temperatures and therefore,was not included in the simulations.

Temperature fields at various times are shown atthe mid-plane of the bar and plate castings inFigure 3. For the bar casting (see Figure 3(a)), thecalculated temperatures are nearly isothermalat t = 10 s. At t = 41 s and t = 60 s, however, chill-

Figure 2. Inputs for the stress analysis included Young’s Modulus (a), the initial yield stress and hardening exponent (b), and thelinear thermal expansion for steel (c).

126 D. GALLES ET AL.

induced spatial temperature gradients that aremostly parallel to the horizontal plane can beseen. It will be shown in the following sectionthat these gradients generated a bending momentin the bar. After only t = 1 min 32 s, however, thegradients are much less prominent, as tempera-tures tend towards an isothermal state. Finally, att = 4 min 6 s, temperature variations can no longerbe seen in the bar. Consequently, minimal bendingof the bar caused by thermal stresses can beexpected after this time.

Due to its larger size, the cooling times in theplate casting are significantly longer than those inthe bar, as shown in Figure 3(b). Regardless, spatialtemperature gradients can again be seen through-out the casting, most notably at t = 5 min and t= 21 min. After these times, the plate temperaturefields are nearly isothermal. Contrary to the barcasting, however, variations in section thicknessthroughout the plate geometry naturally inducedspatial temperature gradients. In particular, therisers are chunky and formed hotspots that cooledslowly. Conversely, the gating cross section is smallin comparison to the other components of the cast-ing system and therefore cooled relatively fast. Thisobservation is notable, because the cooling patterns

in the plate casting will closely resemble those thatcan be expected in a production casting. Thus,Figure 3(b) illustrates how essential components ofa casting system (e.g. gating and risers) inherentlygenerate uneven cooling that can lead todistortions.

The calculated temperature fields were written ata sufficient number of time steps to ensure a smoothtemperature profile at all material points. The resultswere then copied onto the finite element mesh usedfor the stress analysis.

Stress simulations

3D Stress simulations were performed using the generalpurpose finite element code FEAP. In order to preventrigid body translations and rotations, minimal boundaryconditions were specified at the mid-length of bothcastings, which assured symmetric deformations aboutthe casting mid-plane. Due to its larger size, advantagewas taken of the plate symmetry and only ¼ of thegeometry was modelled. First order brick elementswere used for the simulations.

Loop tests were conducted to evaluate the inversesolution. These tests involved performing an inversestress analysis first in order to determine the pattern

Figure 3. Predicted temperatures at the casting mid-plane are shown at various times for the bar (a) and plate (b) castings.

INTERNATIONAL JOURNAL OF CAST METALS RESEARCH 127

geometry, which was then used as the input geometryfor the forward stress analysis. If successful, the forwardanalysis will recover the initial geometry used for theinverse analysis within a reasonable level of accuracy.Both analyses commence from an initial state withoutstresses or plastic deformations and are prescribed thesame loading (i.e. thermal) history. The computationalresources needed to perform the forward and inversesimulations were essentially equal.

In order to quantify differences between the for-ward and inverse analyses, the following metrics wereadopted. A configuration error,err confð Þ, evaluateddifferences between calculated inverse and forwarddisplacements and was computed by normalizing dif-ferences in the current position x by the forwarddisplacement u:

err confð Þ %½ � ¼

ðΩ

xInverse � xForward�� ��dv

ðΩ

uk kdv� 100 (4)

In Equation 4, xInverse refers to the input geometry forthe inverse simulation, whereas xForward relates to thecalculated coordinates of the forward simulation. Inaddition, a scaled error norm,err �ð Þ, was computed forthe equivalent plastic strain and von Mises stressusing the following relation:

err �ð Þ %½ � ¼

ðΩ

Δ �ð Þk kdvðΩ

�ð Þk kdv� 100 (5)

In Equation 5, �ð Þ represents the quantity of interest(i.e. equivalent plastic strain or von Mises stress)

calculated from the forward analysis and Δ �ð Þ is thedifference between the forward and inverse solutions.

In addition to the computed errors, the final verti-cal profile of the casting’s top edge is plotted to helpthe reader visualize the deformations. Also, the tem-poral evolution of vertical displacement for a singlepoint located at the upper left corner of each castingis plotted.

Bar castingThe loop test for the bar casting commenced with aninverse analysis, which calculated the configurationshown in Figure 4(a). Distortions are enhanced 20xto aid visualization. Recall that the input casting geo-metry for the inverse simulation is shown in Figure 1(a). Although it was not included in the stress analysis,the chill induced uneven cooling that led to anupward distortion of the bar ends. Keep in mindthat this deformation was calculated within theinverse framework and is therefore reverted. In reality,the ends of a physical bar produced under the pre-sent conditions will distort downward. This behaviouris easily confirmed through a forward stress analysis.In a similar vein, the usual thermal contractions thatoccur during casting are manifested as expansions forthe inverse problem. For example, the calculated barlength in Figure 4(a) spans from −153.15 mm to153.15 mm, a total of 306.3 mm. This value exceedsthe input geometry length by 6.3 mm and corre-sponds to the unrestrained (i.e. free shrink) patternallowance of 2.1% for a low allow steel. Thus, theinverse technique accounts for both mechanical andthermal strains to determine a pattern shape that canbe expected to produce the as-cast geometry shown

Figure 4. Bar configurations calculated by the inverse (a) and forward (b) simulations. The vertical position along the top edgeof the bars is plotted as a function of axial position in (c). A 20x deformation factor is used in (a) and (b).

128 D. GALLES ET AL.

in Figure 1(a). For the second part of the loop test, thepattern shape determined by the inverse analysis(Figure 4(a)) was used as the input geometry for theforward analysis, which then calculated the shapeshown in Figure 4(b). This configuration appears tobe similar to the original input geometry shown inFigure 1(a) that was inputted into the inverse analysis.

The calculated bar configurations in Figure 4(a) and4(b) were further analysed by plotting the verticalposition of the bars’ top edges as a function of axialposition in Figure 4(c). The reference lines in Figure 4(a) and 4(b) represent the zero vertical position. Forthe inverse analysis (blue curve), the top edge gener-ally increases from an initial constant vertical positionof 25 mm, which corresponds to the input geometryshown in Figure 1(a). This increase is due toa combination of thermal strains and distortions. Thehorizontal dashed line in Figure 4(c) represents thevertical increase of the top edge due to thermalstrains for the inverse analysis. Deviations from thisline represent distortions, which increase monotoni-cally from zero at axial position = 0 to a maximumvalue at the bar ends. The increased bar length calcu-lated by the inverse analysis due to thermal strains isalso apparent in Figure 4(c), as the inverse analysiscurve extends beyond the two vertical dashed linesthat represent the original bar length.

The most important takeaway from Figure 4(c) isthat the configuration calculated by the forward ana-lysis recovers the top edge dimensions of the bargeometry inputted into the inverse analysis.Specifically, the 300 mm bar length and constantvertical profile of 25 mm for the top edge is identicalfor both geometries. Since the deformation for thebar is relatively simple, recovery of the top edge can

be expected to result in recovery for the rest ofthe bar.

The temporal evolution of vertical displacementfor a single point at the upper left corner of the baris plotted on complete (30,000 s) and 200 s timescales in Figure 5(a,b), respectively. These plotsdemonstrate that for all times, the inverse and for-ward evolutions are equal in magnitude but oppo-site in sign. Thus, at any given time step, theforward calculation recovers the original geometryused for the inverse simulation. The forward curveprovides a physical interpretation of the evolution(recall that the inverse simulation produces unphy-sical results). The forward curve begins to decreasedownward at approximately 10 s and reaches a mini-mum value of −2.15 mm at 45 s. This sharp down-ward deflection is due to rapid cooling andsolidification at the bottom of the casting, which isnot only accompanied by an increase in strengthbut also triggers the onset of thermal contraction.Meanwhile, the upper portion of the bar is stillmostly liquid and therefore, relatively weak. Asa result, thermal contractions in the bottom portionof the bar create a bending moment that easilydistorts the bar ends downward while also plasti-cally deforming solidified regions in top of the bar.As the solidification front progresses upward, theupper portion of the bar gains strength and alsobegins to thermally contract. This counteracts thebending moment that caused the initial downwarddistortion. Consequently, the bar ends deflectupward, which is seen as an increase in verticaldisplacement from 45 s to approximately 150 s.After 150 s, small negative changes in forward dis-placement are due to additional thermal strains, as

Figure 5. Vertical deflection at the upper left corner of the bar plotted as a function of time on complete (a) and 200 s (b) timescales. The configuration error is shown in (c).

INTERNATIONAL JOURNAL OF CAST METALS RESEARCH 129

minimal temperature gradients at these later timescannot generate the thermal loading needed toproduce distortions. The final bar shape is charac-terized by a slight downward distortion at its ends,which is manifested as a negative vertical displace-ment at 30,000 s for the forward curve. This finalshape was anticipated due to the generation ofplastic strains at early times that increased thelength at the top of the bar and prevented thebar ends from returning to their original verticalposition. An additional feature of the curves inFigure 5(a) is the small ‘wiggle’ seen at 1000 s,which is attributed to a brief volumetric expansionthat accompanies the decomposition of austenite toferrite and cementite.

The elemental configuration error is shown ona contour map in Figure 5(c). The largest errors areseen at the ends of the bar, which correspond to theregions of largest deformation. However, the maxi-mum configuration error for the bar is less than0.04%. Furthermore, the average configuration errorover the entire bar geometry is 0.01% (see Table 1).Such small differences in configuration can beneglected for all practical purposes. Accordingly, theloop test validates the finite inverse deformation ana-lysis for the bar casting.

Contour plots of von Mises stress and equivalentplastic strain are plotted for the inverse and forwardsimulations and are shown with their associatederrors in Figure 6. Errors are shown on the desiredas-cast configuration. In general, the von Misesstress and equivalent plastic strain contour maps

are similar for both for the forward and inversesimulation. The maximum error values are 6.9%and 1.5% for the von Mises stress and equivalentplastic strain, respectively. However, the averagerespective errors over the entire domain are only0.23% and 0.29% for von Mises stress and equiva-lent plastic strain, as shown in Table 1.

Plate castingThe bar casting in the previous section was shown torecover the desired as-cast shape through successiveinverse and forward stress simulations. Such a simplegeometry, however, is not representative of a typicalfoundry casting. In reality, casting systems containfeatures (e.g. gating, risers) that inherently createcomplex geometries, which in turn may lead touneven cooling that generates distortions. Therefore,the plate casting model in Figure 1(b) was created toprovide a realistic (albeit simplified) casting system towhich the finite element inverse deformation analysiscould be applied.

The inverse analysis was again performed first. It waspreviously shown during the heat transfer analysis thatthe gating and risers naturally created uneven coolingand associated thermal loading for the plate casting sys-tem. Such loading resulted in the calculated configurationshown inFigure 7(a). Since deformations were symmetricabout the vertical plane, only one half of the geometry isshown. In general, the deformation behaviour for theplate resembles that of the bar. Specifically, 1) the endsof the plate are distorted upward and 2) the castinglength increases. The upward distortion is expectedbecause the cooling trends are the same for both cast-ings, i.e. the bottom cools faster than the upper portion ofthe casting. The length increase, as previously discussed,is due to calculation of thermal strains in the inverseframework. In addition to distortions in the plate, thegating is also distorted upward at its mid-length.Despite this additional complexity, the configuration

Table 1. Average errors of configuration, equivalent plasticstrain, and von Mises stress.Case err(conf) err(ep) err(qv)

Bar 0.01% 0.29% 0.23%Plate 1.78% 2.38% 6.56%

Figure 6. Von Mises stresses, equivalent plastic strains, and associated errors for the bar. A 20x deformation factor is used in (a)and (b).

130 D. GALLES ET AL.

calculatedby the forward simulation (see Figure 7(b) onceagain recovers the input geometry that was used for theinverse analysis. The final vertical displacement at the topedge for both configurations is plotted as a function ofaxial position inFigure 7(c). As in the bar example, thermaland plastic strains cause the vertical position of the topedge to evolve from a constant vertical height of 25 mmto the profile shown by the blue curve for the inversesimulation. The forward simulation subsequently recoversthe constant vertical profile of 25 mm, as demonstratedby the red curve in Figure 7(c).

The temporal evolution of vertical displacement fora point at the top left corner of the plate is plotted oncomplete (80,000 s) and 5000 s time scales. Althoughthe time scales and magnitudes of displacement aredifferent, characteristic features of these plots resem-ble those of the bar casting. In particular, the initialdecrease in forward displacement (to a minimumvalue at 675 s) is followed by an increase (untilapproximately 5000 s), after which the decompositionof the austenite phase is accompanied witha volumetric expansion that causes the plate ends todisplace downward (from 5000 s to 8000 s). After thephase transformation is complete, additional thermalcontractions cause the plate ends to gradually dis-place upward until 80,000 s. The configuration errorshown in Figure 8(c) is larger than that for the bar butstill reasonably small (<2%). The average configura-tion error for the entire plate casting system (1.78%) issummarized in Table 1.

Contours of von Mises stress, equivalent plasticstrain, and associated errors are shown in Figure 9.Errors are plotted on the desired as-cast configuration.Although the forward and inverse contours againappear to contain no discernible differences, the

error calculations reveal local differences that areless than 10% and 4% for von Mises stress and equiva-lent plastic strain, respectively. The average errorsover the domain, shown in Table 1, are 6.56% and2.38% for von Mises stress and equivalent plasticstrain, respectively.

In general, the computed errors for the bar weresmaller than those for the plate. A likely explanationfor this difference can be found by comparing the max-imum vertical deflection at the ends of the castings. Theend of the bar and plate castings distort to maximumabsolute values of 2.15 mm (at 45 s) and 5.93 mm (at675 s), respectively (see Figure 5(b) and 8(b)). Therefore,the material points for the plate generally travelleda larger displacement path than those for the bar. Forthis reason, larger errors can be expected in the platecasting. Regardless, the configuration errors for bothcastings are reasonably small, as the finite inverse defor-mation analysis calculated pattern shapes with excellentaccuracy.

A final comment regarding the vertical displacementplots in Figures 5 and 8 is warranted. For this study, thedisplacement evolutions were equal but opposite, asthe forward and inverse curves mirror each other aboutthe horizontal origin. This observation may prompt thereader to suggest that one could simply subtract theforward displacements from the as-cast configurationsin Figure 1 to determine the pattern shapes. Sucha strategy may indeed predict the pattern shape withgood accuracy, provided that the displacements arevery small. In this case, the forward displacementwould be nearly the same whether it was determinedusing either the as-cast or pattern configuration.However, if the displacements are moderately large,the as-cast and pattern configurations may differ

Figure 7. Plate configurations calculated by the inverse (a) and forward (b) simulations. The vertical position along the top edgeof the plates is plotted as a function of axial position in (c). A 10x deformation factor is used in (a) and (b).

INTERNATIONAL JOURNAL OF CAST METALS RESEARCH 131

considerably, for which case the forward displacementsfrom the two configurations will not match each other.In contrast, the inverse and forward displacements thatwere determined from their respective as-cast and pat-tern configurations will always mirror each other. Forthis reason, the inverse deformation analysis is prefer-able, as it provides a robust and general method foraccurate calculation of pattern dimensions over a widerange of deformations.

Conclusions

Casting distortions arise during metalcasting and cre-ate dimensional inaccuracies in the as-cast part. Theirpresence necessitates a costly trial-and-error patterndesign process that extends lead times and reducesefficiency. The present study explores the viability ofthe finite element inverse deformation analysis tocalculate pattern allowances for steel castings. The

Figure 9. Von Mises stress, equivalent plastic strains, and associated errors for the plate casting. A 10x deformation factor is usedin (a) and (b).

Figure 8. Vertical deflection at the upper left corner of the plate plotted as a function of time on complete (a) and 5000 s (b)time scales. The configuration error is shown in (c).

132 D. GALLES ET AL.

underlying concept behind this technique involvesfinding, by solving a boundary value problem,a deformation that is the kinematic inverse of thatfor the usual forward analysis. As a result, the inverseanalysis essentially works backward, starting from thedesired as-cast configuration, to determine patterndimensions in a single design iteration.

The inverse deformation analysis was evaluatedthrough a loop test, which involved performing aninverse simulation first using the desired as-cast con-figuration as input. The resulting calculated patternshape was then inputted into a forward analysis,which attempted to recover the original geometryused for the inverse analysis. An important caveatfor this recovery concerns the elastoplastic constitu-tive relation used in this study. The calculation ofplastic deformations will generally lead to differentloading paths between the forward and inverse pro-blems. However, since plastic strains (i.e. distortions)for casting processes are typically on the order ofa few percent, only marginal differences wereexpected between the inverse and forward loadingpaths. This was indeed the case for the plate castingsystem, which contained a moderately complex geo-metry and deformation history. Nonetheless, the looptest accurately recovered the original geometry, asthe average difference between inverse and forwardconfigurations was less than 2%.

The finite element inverse deformation analysispotentially provides an efficient means to calculatepattern dimensions. Although this study representsan important step towards achieving this goal,further work is needed before this technique canbe applied to production castings. Primarily,mechanical interactions between the casting andmould must be considered. Currently, the numericaldifficulties associated with contact interactionsprompted the authors to only consider distortionscaused by thermal stresses. Overcoming this issue isparamount to the success of the inverse deforma-tion analysis. It should also be noted that accuracyof the inverse analysis was evaluated based oncomparisons to the forward analysis. Thus, theimportance of the forward simulation’s predictivecapability cannot be overstated. The present mate-rial model is based one that was previously cali-brated for a low alloy steel using in situ data fromcasting experiments. If the finite element inversedeformation analysis is used for other casting mate-rials, care must be taken to properly calibrate theassociated material model parameters for the for-ward problem. Despite these concerns, the presentresults lend confidence to the inverse deformationanalysis for its ability to ultimately determine pat-tern allowances accurately and with unprecedentedefficiency.

A final comment regarding the applicability of thepresent computational technique to other casting pro-cesses is warranted. In this study, the finite elementinverse deformation analysis was utilized for steel sandcastings. However, this is a general method that can alsobe applied to other casting processes (e.g. permanentmould, investment, continuous) as well as heat treat-ments for which distortions create dimensionalinaccuracies.

Acknowledgements

The views and conclusions contained in this document arethose of the authors and should not be interpreted asrepresenting the official policies, either expressed orimplied, of the Army Research Laboratory or the U.S.Government. The U.S. Government is authorized to repro-duce and distribute reprints for Government purposes not-withstanding any copyright notation herein.

Disclosure statement

No potential conflict of interest was reported by theauthors.

Funding

Research was sponsored by the Army Research Laboratoryand was accomplished under Cooperative AgreementNumber W911NF-18-2-0161.

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