We implement an approach using Bayesian inference and machine learni ng to calibrate the material parameters of a constitutive model for the supe relastic deformation of NiTi shape memory alloy. We use a diamond-shaped specimen geometry that is suited to calibrate both tensile and compressive mate rial pa- rameters from a single test. We adopt the Bayesian inference calibrat ion scheme to take full-ﬁeld surface strain measurements obtained using digit al image corre- lation together with global load data as an input for calibration. The calibrati on is performed by comparing these two experimental quantities of int erest with the corresponding results from a simulation library built with the superelastic forward ﬁnite element model. We present a machine learning based ap proach to enrich the simulation library using a surrogate model. This impr oves the calibration accuracy to the extent permitted by the accuracy of the un derlying materialmodelandalsoimprovesthecomputationaleﬃciency. Wedemon strate, verify, and partially validate the calibration results through various e xamples. We also demonstrate how the uncertainty in the calibrated superelas tic material parameters can propagate to a subsequent simulation of fatigue loading. This approach is versatile and can be used to calibrate other models of supere lastic deformation from data obtained using various modalities. This probabili stic calibration approach can become an integral part of a framework to assess and communicate the credibility of simulations performed in the design of supere- lastic NiTi articles such as medical devices. The knowledge obtained from this calibration approach is most eﬀective when the limitations of the under lying model and the suitability of the training data used to calibrate the m odel are understood and communicated. ∗Corresponding author Email address: harshad.paranjape@confluentmedical.com (Harshad M. Paranjape) Preprint submitted to engrXiv August 28, 2020 Keywords: NiTi, Finite element modeling, Uncertainty Quantiﬁcation, Digital Image Correlation, Bayesian Inference 1. Introduction Simulation of the thermo-mechanical response of Nickel-Titanium (NiTi ) shape memory alloys (SMAs) remains a topic of signiﬁcant interest in th e sci- entiﬁc community and in engineering practice. The reason for scient iﬁc interest in NiTi simulation is due to the challenges posed by the multi-scale m icrostruc- ture, highly non-linear and inelastic constitutive response, anisot ropy of the material properties, and the asymmetry in tension vs. compression re sponse. The engineering interest in NiTi simulation is due to the extensi ve use of su- perelastic NiTi in the manufacturing of medical implants and devices and due to the strong potential of shape memory NiTi as an actuator material in the aeronautics industry. When computational methods are used to simulat e NiTi response in these ﬁelds, a model is typically used to predict ei ther global quanti- ties of interest such as the radial force of a stent or the actuation load of a s pring actuator, or to obtain local quantities of interest such as the local strai n distri- bution in an implant under physiological boundary conditions. These quan tities of interest are then leveraged for some speciﬁc context of use (e.g., as p art of the performance or durability assessment). As with the broader simulation community, a topic of emerging importance to NiTi simulation is the necessity to quantify and report the credi bility of any computational modeling that is performed for a speciﬁc context of use [ 1]. Or- ganizations in both medical device and aeronautics communities recognize that simulation will play an increasingly prominent role in decision makin g through- out the lifecycle of a particular product. They identify credibi lity assessment of the models and uncertainty quantiﬁcation of the simulation results as a key component of the future development in the simulation practices. Mor rison et al. note that suﬃciently-credible computational modeling evidenc e can act as a support in the regulatory applications of medical devices [2]. NASA Vision 2040 report speciﬁcally incorporates uncertainty quantiﬁcation as a key f ocus area [3]. The simulated response from any computational model is inﬂuenced by t he material parameters used as constitutive model inputs. As a conseque nce, the credibility of simulation results is aﬀected by the speciﬁc value s and the uncer- tainty of the material parameters that are used. The material parameters f or a particular computational model are generally calibrated by comparing ex per- imental measurements (e.g., load or strain) with computational simulati ons of the experiment. While ad hocor trial-and-error approaches are the most com- mon, recently, various eﬀorts have employed statistical or probabil istic methods such as Bayesian inference (BI) to solve the inverse problem of cali brating the material parameters [4]. The beneﬁt of probabilistic methods over de terminis- tic optimizations schemes for calibration (e.g., ﬁnite element mode l updating, gap method, virtual ﬁelds method) is that the probabilistic method s inherently 2 furnish uncertainty information about the ﬁtted parameters, rather t han simply providing the optimum parameters. Gogu et al. presented a BI approach f or thedeterminationofmacroscopicelasticstiﬀnessoftruss-typemet allicstructure [5]. Liu and Au used BI to determine the material parameters of a phenomen o- logical elasto-plastic model for the hysteretic load-displacement re sponse of a composite material [6]. Castillo and Kalidindi reported a calibration me thod based on BI to determine the single crystal elastic constants of metall ic materi- als using indentation load-displacement data [7]. Ricciardi et al. dem onstrated the calibration of a crystal plasticity model using a BI approach [8]. Tw o eﬀorts in the literature have speciﬁcally applied BI to the calibration of Ni Ti SMA con- stitutive model properties. Crews and Smith used BI to estimate the material parameters of a phenomenological model for actuation response of NiTi due to the shape memory eﬀect [9]. Honarmandi et al. presented another example of calibration of a model for shape memory eﬀect in NiTi [10]. These two examp les addressed the calibration of models to simulate the thermally-indu ced response of NiTi and not the superelastic response. A common aspect to the example s cited above is that they used a single experimental comparator (e.g., gl obal load data) to arrive at the calibrated material parameters. Generally, there is a three-fold reason to perform calibration using a rela- tively large comparator data set. First, the uncertainty in the calibr ated ma- terial properties reduces as the experimental sample size increas es. Second, a larger sample size reduces the eﬀect of any prior assumptions made relat ed to the distribution of material parameters. In the context of BI, this ph enomenon issometimesreferredtoasthe data overwhelming the prior . Third, amodelcali- bration performed using a single comparator such as the global load may furn ish material parameters that are optimized to give the correct load response. How- ever, whenthesamemodelisusedinanothercontextofuse(e.g., tos imulatethe local strains), then the predictions using the optimum parameters may signiﬁ- cantly deviate from the expected response in terms of the quantity of interest. The ﬁrst two concerns noted here can be simply addressed by perfor ming a large number of experiments and using a larger number of observations to per form the calibration. However, the third concern can be alleviated by usin g a variety of comparators obtained using diﬀerent experimental modalities to pe rform the calibration. One approach to obtain a large quantity of comparator data other than load is to use full-ﬁeld surface strain measurements on the test samples and using the strain ﬁeld in the calibration process [11]. Digital image corr elation (DIC) is a versatile method to obtain full-ﬁeld surface strain data during tensile testing of materials. Rethore presented a general strategy for identi fying mate- rial parameters from full-ﬁeld displacement data [12]. Bertin et al. p resented an example of a DIC-based constitutive model calibration approach where they developed a method to determine crystal plasticity parameters fr om small-scale DIC measurements [13]. While it is appealing to use BI on large comparator sample sizes or with comparator data from multiple sources, there is a practical limitation. Mate- rial property calibration using BI is typically implemented using a sampling method such as Markov Chain Monte Carlo. A reliable distribution of the cali- 3 brated material parameters is obtained by performing this sampling th ousands of times with varying material parameters and comparing the prediction s of the forward simulation in each step with the appropriate comparator. Performi ng simulations on-line during the sampling process is quite impract ical, particu- larly when the constitutive response is highly non-linear such as in the case of NiTi alloys. A more tractable solution to this problem is to create a lib rary of simulations beforehand with the input parameters spanning a reasonab le pa- rameter space. Then, during each sampling step this library can be q ueried to obtain the simulation results for the closest available material param eters in the library. This approach may work well when a small library is adequat e. However, if a large number of parameters need to be calibrated, then th e size of the library needed can be quite large. Alternately, this problem can be ad- dressed by developing a surrogate forward model that can eﬃciently fu rnish the simulation results for any input material parameters. Recently, a fe w eﬀorts have used machine learning (ML) to develop a surrogate model to speed up ma- terial model calibration using BI. Wu et al. used neural networks to s peed-up the calibration of a homogenized elasto-plastic model for composites [14]. Lu et al. used deep learning to determine elasto-plastic material prope rties using indentation comparator data [15]. However, they did not use BI to determi ne parameter calibration uncertainties. In summary, BI is a useful approach to determine constitutive mode l param- eters and associated uncertainties. It is desirable to use multi- modal experimen- tal data such as full-ﬁeld strain data together with load data as a comparator. ML can be used to accelerate the BI parameter determination through the gen- eration of a regression model that acts as a surrogate to the original constitu tive model. To our knowledge, there is not an eﬀort in the literature consi sting of these three components to calibrate inelastic models in general and p articularly the computational models for superelastic NiTi. Thus, we address thi s gap by implementing a combined BI and ML approach to calibrate and optimize su - perelastic NiTi constitutive model parameters from load and full-ﬁel d surface strain data. We demonstrate this calibration approach with the phenomenol ogi- cal superelastic constitutive law implemented in Abaqus ﬁnite el ement modeling (FEM) framework [16]. However, the framework presented here is gener al and can be applied to other models for superelasticity. In Section 2, we d escribe the methods used in this calibration framework. In Section 3, we ﬁrst present a basic example of the BI calibration approach when the load-displacement d ata from a tension test of NiTi is used as a comparator. We then present a de- tailed example demonstrating all elements of the proposed approach. Fin ally, we provide an application of this calibration approach. We demonstrate how the uncertainty in the calibrated material parameters obtained using BI can be propagated to subsequent simulations performed using these material proper- ties. In Section 4, we critically assess the advantages and limitations of our approach. 4 2. Calibration Framework We will ﬁrst describe the calibration procedure in general. Later, w e will specialize the procedure to the speciﬁc phenomenological model for superelastic deformation of NiTi. Consider a constitutive law of the form σ(ε,m) whereσis the stress,εis the strain, and mis a material parameter vector. Let Q(σ,a) be a vector of Quantities of Interest (QoI) that can be determined for a speciﬁc deformation geometry and boundary conditions. Here ais a parameter vector encoding the detailssuchasgeometryandboundaryconditionsthatarenecessarytocalc ulate Q. For example, Qcould be the global load measured during the deformation of a specimen at a ﬁxed imposed displacement. Or Qcould be the strain tensor at a speciﬁc location in the specimen at an imposed deformation. Since Qis based on the constitutive law, it indirectly depends on m. The QoI can either be measured experimentally ( Qexpt) or computationally predicted ( Qsim) using a model or numerical implementation of the constitutive law σ(ε,m). The simplest procedure to calibrate a model of σ(ε,m) (i.e., to determine parameters mcal) is to minimize the diﬀerence between QexptandQsimwhile varyingm. A typical method to solve this inverse problem is to perform the minimization: mcal LSQ:= argmin mN/summationdisplay iwi/parenleftbig Qsim i−Qexpt i/parenrightbig2. (1) HereNis the total number of observations of Qandwiis the weight assigned to each observation. The objective function on the right hand side has a quad ratic form and thus, we refer to this procedure as least-squares calibration . While this procedure is straight-forward, it does not furnish any informat ion about the uncertainty of the calibration. BI on the other hand, provides a to ol to determine mcaland quantify the uncertainty in the calibration. The general scheme of our calibration method is shown in Figure 1. In the ﬁrst step of this calibration procedure, we standardize a test spec imen geometry and a test protocol. Based on these, we can experimentally obtain a numb er ofQexptvalues. In parallel, we setup a simulation with the constitutive la w σ(ε,m), the virtual specimen geometry, and boundary conditions based on the test protocol. The simulation acts as a forward model and furnishes a nu mber ofQsimvalues. In the simulation, we vary the values of the material parameters mto span the typical material parameter space and obtain the corresponding Qsimvalues. Thus, we build a simulation library that maps mtoQsim. In the next step, we use BI to obtain mcaland their uncertainty. We ﬁrst describe the BI component of this procedure below. 2.1. Bayesian Inference for Calibration and Uncertainty Quantiﬁ cation The probability distribution of the material parameters given a set of e x- perimentally measured quantities of interest ( P(m|Qexpt) or posterior) can be expressed in terms of the probability distribution of the material p arameters 5 Figure 1: A summary of the Bayesian Inference material param eter calibration method. A ﬂowchart listing the key components of the method is shown. based on prior knowledge ( P(m) or prior) and the probability of observing an experimental response Qexptif the material parameters were actually m (P(Qexpt|m) or likelihood) using the Bayes’ theorem P(m|Qexpt) =P(m)P(Qexpt|m) P(Qexpt). (2) The denominator, P(Qexpt), is a normalization constant and referred to as the probability of the evidence [17]. In BI, this factor is typically not cal culated and simply P(m|Qexpt)∝P(m)P(Qexpt|m). Bayes’ theorem in the form described above furnishes a probabilit y distribu- tion of the material parameters given certain experimental data. The opti mum or ﬁtted parameters can be reported from this distribution in terms of one of the point estimates. Some common point estimates are mean, median, or maximum a posteriori (MAP). The uncertainty in the ﬁtted parameters can be re ported in terms of the credible intervals of the distribution. Credible intervals are the probabilistic statistics counterpart to the conﬁdence intervals i n the frequentist statistics. Thus, uncertainty estimation of calibration is built in to this method. The prior P(m) incorporates an expert’s knowledge about the parameter space for a particular constitutive law. For example, the Young’s modulus of most metals is in the GPa range with a larger probability between 10 to 100GPa. The likelihood ( P(Qexpt|m)) of observing a particular experimental response for a given material parameters can be calculated using a forward model th at furnishes Qas a function of m. A simulation library that maps mtoQsim 6 across the parameter space can be used to calculate the likelihood. Th us, P(Qexpt|m) :=c·N/productdisplay iexp/parenleftBigg −wi(Qsim i−Qexpt i)2 2s2/parenrightBigg , (3) wherecis a constant, ware the weights assigned to the observations, and s2is the standard deviation of the lumped error in the experimental measur ements [4]. The exponent in this expression is a loss function that quantiﬁe s the dis- crepancy between the experimental measurement and a trueresponse observed for a particular mand reaches a maximum value when the discrepancy between the observed and the true response is minimum [18]. If multiple ind ependent ex- perimental observations are available and the errors in the observations (s2) are assumed to be constant, the total likelihood can be modeled as a multi variate Gaussian distribution, resulting in the product term in Equation (3). In most practical situations, no analytical expression for any of the term s on the right hand side of Equation (2) is available. Hence, a numerical sampl ing procedure is generally used to indirectly obtain the posterior dis tribution of the material properties from the prior distribution and the likelihood function. In this calibration scheme, we use a Metropolis-Hastings variant of the Mar kov Chain Monte Carlo (MCMC) sampler to obtain the ﬁtted parameter distri bu- tion. An MCMC sampler operates in three steps [4, 17, 19]. First, the mater ial parameters are initiated at a certain point in the parameter space ( m) and the posterior probability at that point is calculated knowing the likelih ood function and the prior distribution. Second, a new point miis drawn in the material parameter space and the posterior probability at the new point is calculat ed. In the third step, the new sample is accepted if the new probabilit y is larger than the previous. However, the new sample is also accepted if the acc eptance probability is larger than a random number drawn from the uniform standar d distribution. Steps two and three are repeated until a certain num ber of samples are obtained. From this sampled posterior distribution, the point est imates and the credibility interval described above can be readily calculate d to obtain the optimum material parameters and their uncertainty. There is one challenge in executing the approach described so far. On e of the terms in the likelihood function in Equation (3) is Qsim. If the forward model used to calculate Qsimfrom a sampled mis fairly complex, then it may be challenging to use the model online during the MCMC sampling. On line cal- culation of Qsimmay be avoided by building a library of Qsimbeforehand and then querying the library during the sampling process for the cl osest available material parameters to mi. This is akin to doing a nearest-neighbor interpo- lation on the simulation results in the parameter space. A superior appr oach is to develop a computationally eﬃcient surrogate model that is derive d from the available simulation results to furnish Qsimfor anym. We describe the development of such a surrogate forward model below. 7 2.2. Surrogate Model from the Simulation Library using Machine L earning A surrogate model that can furnish Qsimfor anymcan be built using any regression scheme if a library of previously run simulations that map m→Qsim is available. In this work, we choose a ML regression method – kernel support vectormachine(SVM)–tobuildthesurrogatemodel[20]. Oneregression model istrainedforeachQoIthatwillbeusedformaterialparametercalibration . Once a regression model is trained, it can be used in the MCMC sampler des cribed above to obtain the likelihood P(Qexpt|m) formthat is not present in the simulation library. Figure 2: Test specimens and the quantities of interest (QoI s). (a) Stress-strain curve of a NiTi dogbone specimen tested per ASTM 2516. The specimen geo metry is shown on the left inside the inset. The single element simulation geometry us ed to model the tensile response of this specimen is shown on the right inside the inset. (b) A d iamond NiTi specimen load- displacement response during an isothermal tensile test. T he specimen geometry is shown inside the inset. The simulation geometry consists of a quar ter diamond and is highlighted in orange in the inset. (c) A typical surface strain ﬁeld in a NiT i diamond specimen obtained using a digital image correlation (DIC) measurement during a tensile test. In all cases, the loading direction is along Y. 2.3. BI and ML Calibration Approach for a Superelastic Material Mo del Now we specialize the BI- and ML-based calibration approach to a phe- nomenological model for superelastic deformation behavior. Figure 2(a) sh ows the constitutive response of a typical superelastic NiTi specimen at room tem- perature. The model of Auricchio and Taylor is widely used to simul ate such deformation response [16, 21]. The model is developed in an inﬁnitesimal strain framework. Stress and temperature are considered as the independen t variables. The inherent deformation mechanism of austenite to martensite phase transfor- mation is simulated. The total martensite phase fraction at a material poin t is taken as the internal variable. The total strain at a material point is addit ively decomposed into an elastic and a transformation component ( ε=εe+εtr). The elasticbehavioroftheausteniteandmartensitephaseisassumedtob eisotropic. The evolution of the internal variable is determined using a rate-in dependent formulation. This model is implemented in various commercial FEM pac kages includingAbaqus(SimuliaDassaultSystemes, version2019). Themode laccepts six key material inputs: Young’s modulus of austenite ( EA), Young’s modulus of martensite ( EM), maximum transformation strain ( εt), upper plateau stress 8 in tension ( σUPS), lower plateau stress in tension ( σLPS), and the compression plateau stress ( σCPS). The connection of these six parameters to the superelas- tic constitutive response is schematically shown in Figure 2(a). Wh ileσCPSis dependent on the microstructure and the processing history of a sp eciﬁc NiTi specimen, often σCPSis not calibrated in simulations and it is assumed that the ratio σCPS/σUPSapproximately equals 1.5 [22, 23]. We do not make this assumption. The model allows to specify the hardening of upper and l ower plateaus. We assume a ﬁxed hardening of 30MPa. The Abaqus implementa- tion of the model takes other minor inputs. Those are not addressed in th is work. Plasticity is also not addressed. In summary, the goal of this BI- an d ML-based calibration approach is to solve the inverse problem of determi ning m={EA,EM,εt,σUPS,σLPS,σCPS}given a set of experimental inputs. 2.3.1. Calibration Using Global Stress Data As a ﬁrst example of the BI-based calibration approach described in Sec- tion 2.1, we calibrate the superelastic material model using the obser vations from a tensile test on a dogbone specimen that was laser-cut from NiTi tub ing. We heat treated a commercial tubing material with 50.8at.%Ni composition such that it was superelastic at room temperature. The dogbone test spe cimen geometry is shown in Figure 2(a) inset. An experimental stress-strai n curve obtained at room temperature according to the standard test method in AS TM F2516 [24] is shown in Figure 2(a). The test method consists of loading the specimen in displacement control to an engineering strain of 6%, unload ing to zero load, and ﬁnally loading to fracture. We performed the test on an Ins tron 5969 load frame at a nominal strain rate of 1 .4×10−4s−1. The axial strain in the gage was measured using the built-in video extensometer in th e Instron load frame. We calculated the axial stress in the specimen gage from the load measured by a 1kN load cell on the Instron load frame and the cross-section dimensions of the test specimen measured prior to test. We extract ed axial true stress at 82 points along the stress-strain curve such that the points were equally separated in time. These axial stress values will serve as Qexptin the material parameter calibration. We created a single element Abaqus/Standard FEM simulation of the load- ingconditionusedintheexperiment. Thedeformationinthegageofadogb one- type NiTi specimen can be adequately captured by a single element mo del. We modeledtheresponseofthesingleelementmodelusingthesuper elasticmaterial model in Abaqus. We created a library of 1843 instances of the simulation wi th the material parameters in each case selected using Latin hypercube s ampling of the following parameter space: EA∈[10GPa,80GPa], EM∈[10GPa,50GPa], εt∈[0.03,0.07],σUPS∈[100MPa ,600MPa], σLPS∈[10MPa,400MPa], and σCPS∈[150MPa ,700MPa]. We started with smaller library sizes and eventu- ally settled with 1843 simulations since they furnished reasonable re sults for this example. Unphysical material parameter combinations such as σUPS≤σLPS were not instantiated. We applied boundary conditions that were equiv alent to the experimental boundary conditions used in the dogbone tensile t est de- scribed above. From the result of each simulation, we extracted axial t rue stress 9 at 82 frames that were equally separated in time. We selected the time points such that the simulation axial stress data was available at the same time i n- stants at which the axial stress from the experiment was extracted. T his is rather straightforward to do considering that both the experiment and simula- tion were executed in displacement control with a constant strain rat e boundary condition. These axial stress values will serve as Qsimin the material parameter calibration. From the experimental stress-strain data for the dogbone and the simul a- tion results in the library, we performed material property calibr ation. All data analysis was performed in Matlab (The Mathworks, Inc., version R2018a). We selected global axial true stress at 82 locations along the stress-strain curve as the QoI for this BI calibration example. From these QoI, we ﬁrst perfor med a traditional weighted least-squares calibration to obtain the optimum val ues of the six material parameters. For this, we performed minimization as sp eciﬁed in Equation (1) using a weight at each point that was proportional to the local slope of the experimental stress-strain curve. The speciﬁc weigh ts used at 82 QoIs are listed in Supplementary Data. Weights proportional to the local slope of the stress-strain curve appear to equalize the inﬂuence of data poi nts from the plateaus vs. the points from the elastic regimes. Note that such minimi zation provides the optimum parameters among the parameter combinations in th e 1843 simulation instances. Then, we performed a BI calibration using a G ood- man and Weare aﬃne invariant ensemble MCMC sampler [25]. We used 100 walkers and drew 4 ×106samples. We used an uninformative ﬂat prior. That is, each of the six material parameters had a uniform probability distri bution in the parameter space described in the paragraph above. In BI, the choice of prior can inﬂuence the results. We performed a limited study and found ou t that the results in this case are relatively insensitive to the choice of pr ior distribution. The details of prior eﬀect are provided in Supplementary Data. We ini tialized the walkers with material parameters drawn from a normal distribution cen- tered at the mean of the parameter space and with a standard deviation equal to 1/10th of the parameter space span. We deﬁned the likelihood function as described in Equation (3). We discarded the ﬁrst 50% of the samples from the posterior as the burn-in period. From the marginal posterior distribu tion for each material parameter, we calculated MAP, median, and 95% credible inte r- vals. For this example, we did not perform simulation library enrichm ent using the ML approach described in Section 2.2 since we felt the library si ze of 1843 adequate to demonstrate the BI calibration approach. For reference, the superelastic model parameters for this NiTi tub ing ma- terial, determined using a semi-supervised trial-and-error app roach, are - m= {31 GPa, 26 GPa, 0.045, 340 MPa, 70 MPa, 510 MPa }. 2.3.2. Calibration Using Global Load and Full-ﬁeld Strain Data As a second example of the BI- and ML-based approach, we calibrate the superelastic material model from global load and local strain observations ob - tained during a tensile test of a NiTi diamond specimen laser cut from a strip material with a composition of 50.8at.%Ni. The specimen geometry is shown 10 as an inset in Figure 2(b). We heat treated the NiTi strip material such t hat it was superelastic at room temperature. While the NiTi material used in this example has an identical composition as the tubing material in the previ ous example, the two materials have undergone diﬀerent processing st eps. Thus, the mechanical properties of this material in strip form can be diﬀere nt than those of the NiTi material in tubing form described in the previous se ction. Prior to the test, we applied a speckle pattern of ﬁnely ground charcoal on a matte white background to one surface of the diamond [26]. The experiment al global load-displacement curve for this specimen is shown in Figure 2( b). We performed the test on an Instron 5969 load frame in displacement control at a nominal displacement rate of 2 .94×10−3mms−1at room temperature. During the tension test, we acquired photographs of the specimen gage (i.e., th e curved struts on one side of the diamond) at a regular interval of 1s using a Mitakon Zhongyi 20 mm f/2 4.5X Super Macro lens mounted on a Sony A7III digital mir- rorless camera with 0.2 s exposure and ISO 400. We analyzed the photographs using Ncorr DIC software [27] implemented in Matlab to obtain the full-ﬁ eld 2D surface strain tensor. We used a subset radius of 30 pixels, subset spacing of 2 pixels, and strain window radius of 15 pixels in Ncorr. A sample full -ﬁeld strain map (shear strain component) obtained using this DIC measureme nt is shown in Figure 2(c). We simultaneously recorded the global load-displ acement data during the tensile test using the load cell and the crosshead mo vement respectively on the Instron load frame. Based on the diamond specimen geometry and the experimental test bound - ary conditions described above, we created a simulation of the test. T he dia- mondgeometryhasfour-foldsymmetryandthus, aquarterdiamondasshown in Figure 2(b) inset was modeled to obtain the full deformation response. We mod- eled the response using the superelastic material model in Abaqus. We created a library of 544 instances of the simulation with the material parameters i n each case selected using Latin hypercube sampling of the following parame ter space: EA∈[40GPa,80GPa], EM∈[20GPa,50GPa], εt∈[0.035,0.055],σUPS∈ [300MPa ,500MPa], σLPS∈[100MPa ,300MPa],and σCPS∈[350MPa ,700MPa]. Similar to the ﬁrst example, the library size was determined by tr ial and error and unphysical material parameter combinations were not instantiated. TheQoIforthecalibrationschemearethegloballoadat11equallyseparated points between A and B in Figure 2(b), the global load at 11 equally separated points between B and C in Figure 2(b), and the local mean Green-Lagrange shear strain ( εxy) at four regions in the specimen gage marked by 1 to4 in Figure 2(c) at the same 22 points during the loading cycle where the global load data was extracted. The rationale behind using the shear strain for cali bration is provided in the Discussion section. Thus, 22 global load values and 88 l ocal shear strain values were used as QoI for a total of 110 quantities. We extract ed theQexptfrom the load-displacement and DIC data obtained in the tensile test. The same quantities from each simulation in the simulation libr ary were extracted using Python post-processing scripts. FromQexptandQsimfrom each of the simulation in the simulation library, we obtained the least-squares calibrated material parameter vector by p erform- 11 ing the minimization in Equation (1). We used a weight of 0.25 for the strai n QoI and a weight of 1 for load QoI. These weights and the fact that there are four times as many strain quantities as load ensure that the inﬂuence of l ocal strain and global load on the calibration is equal. Subsequently, we perfor med a BI calibration using the same MCMC sampler as above. During MCMC sam- pling, we selected the simulation result with the nearest availabl emto themi in the MCMC draw to calculate the likelihood function value per th e expression in Equation (3). We did this because Qsimvalues are only available for those mvectors present in the simulation library and not for arbitrary m. The na- ture of the prior and the initial walker values are similar to the exampl e above. We generated 1 ×106samples and rejected the ﬁrst 20% of the samples as the burn-in samples. From the sampled posterior distribution of m, we calculated median and MAP material parameters as well as the credible interval. We r efer to these results as the results from BI approach . To enrich the simulation library, we trained a regression SVM model t o each of the 110 QoI using the results of all 544 simulation instances. Thus, a ﬁt ted SVM surrogate model furnishes the value of that QoI for an arbitrary mgiven as an input. We used the fitrsvm function in Matlab for this purpose. We used a Gaussian kernel with automatic scaling. We standardized the mdur- ing ﬁtting since various material parameters are of widely diﬀerent m agnitudes. We used the automatic optimization option in the ﬁtting function. This option internally ﬁnds such hyperparameters of the SVM that minimize the ﬁ ve-fold cross-validation loss and improve accuracy of the regression. The detail s of veriﬁcation performed on the trained models are described in Suppl ementary Data. The veriﬁcation study demonstrates that 544 simulation samples are ad - equate to furnish a reasonably accurate surrogate model in the six-dim ensional parameter space. Using the trained SVM surrogate models, we performed another calibration using BI. During the MCMC sampling for this calibration, the likel ihood func- tion in Equation (3) was calculated using the QoI values obtained from the trained models. A trained model can furnish Qsimfor an arbitrary m. From the results of this calibration, we again computed descriptors such as m edian and MAP. We refer to these results as the results from BI + ML approach . In summary, we calculated three calibration results for this example: l east-squares, BI approach, and BI + ML approach. One motivation behind using the local strains as a QoI was the belief th at they will furnish such calibrated material parameters that result i n a more accurate simulation of the local strains. To determine if that is the c ase, we also performed a BI + ML calibration where the weights of the strain QoI were set to zero when calculating the likelihood function. Thus, only load QoI w ere used in this calibration. From the sampled marginal posterior probability distr ibutions in this example, we calculated the median calibrated material propert ies. To assess the robustness of the calibration, we performed a simple val idation test. For the validation, we laser-cut a dogbone shaped specimen from th e same NiTi strip material as that used to make the diamond described above. Th is dogbone specimen is diﬀerent from the dogbone specimen laser cut from Ni Ti 12 tubing in the previous example. We performed a tensile test on the dogbone at room temperature. We created a simulation representative of this t est and instantiated the simulation with median, MAP, and 95% credible interv al pa- rameters described above. We compared the simulation results with the tensile test measurements for the purpose of validation. A robust validation is t ypically performedonamodelofequalorhighercomplexitycomparedtothemodel from which calibration results are obtained. Such study is not within the scope of this demonstration. For reference, the superelastic model parameters for this NiTi str ip material, determined using a semi-supervised trial-and-error approach, are -m={20 GPa, 15 GPa, 0.04, 330 MPa, 180 MPa, 495 MPa }. 3. Results 3.1. Calibration Using Global Stress Data The results of least-squares and BI calibration for the example when th e global stress was used as the QoI are listed in Table 1 and graphically shown in Figure 3. The results of least-squares calibration in the table reve al material parameter values in the general regime that is expected for a typical su pere- lastic NiTi tubing material. However, EA< EM. This is unusual but not completely unexpected. Heat treated NiTi typically exhibits an int ermediate R-phase formation during the austenite to martensite phase transform ation. The intermediate phase transformation has a small transformation strain as so- ciated with it and it generally manifests as a softer austenite modulu s in the constitutive response [28]. The marginal posterior probability distri bution plots from BI calibration are shown in Figure 3(a). A narrow posterior probabilit y distribution reﬂects a lower uncertainty in the calibration of that parameter. Thus,EMandεtare calibrated with a lower uncertainty compared to the other four parameters. The lower uncertainty of these two calibrated parame ters is also reﬂected in tighter bounds on the 95% credible interval listed i n Table 1. The calibration results are graphically shown in Figure 3(b). Overall, the MAP point estimate of the posterior distribution shows a closer match w ith the experimental data as emphasized in the inset. The closeness of the m atch can be quantiﬁed and reported in terms of the mean absolute percent error ( MAPE) deﬁned as, MAPE =1 NN/summationdisplay i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleQsim i−Qexpt i Qexpt i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (4) HereNis the total number of QoIs, which is 82 in this example. In the calcula- tionofallMAPEvaluesinthiswork, wedidnotconsidertheQoIscorrespon ding to the ﬁrst two steps of the loading sequence in each experiment. T his is be- cause the load and the strains are close to zero in the ﬁrst two steps of a tensile test and the errors at those steps are of lower concern. MAPE is tabulated in Table 1 and the smallest value is obtained for the MAP point estimate from BI calibration. This suggests that the MAP estimate furnished the most accu- rate parameter calibration. Overall the envelope of stress-strain res ponse from 13 the 95% credible interval is broad. The calibrated and the experiment al stress- strain curves diverge in the large strain regime. This is because plas ticity is not incorporated in the simulations. Fit description EaEmεtσUPSσLPSσCPSMAPE (GPa) (GPa) (MPa) (MPa) (MPa) (%) Least squares ﬁt 26.2 32.7 0.0517 355 61.9 448 15.2 Bi ﬁt (median) 25.0 33.0 0.0519 278 62.6 504 21.6 BI ﬁt (MAP) 28.0 32.6 0.0513 342 55.4 482 12.9 BI ﬁt (95% credible interval)11.6 25.8 0.0440 111 13.0 416 33.6 38.1 0.0595 387 124 665 Table 1: Results of calibration with global stress as the qua ntity of interest. MAPE refers to mean absolute percent error. Figure 3: Results of calibration with global stress as the qu antity of interest. (a) Posterior probability distribution of the six NiTi material model par ameters sampled using the Markov- chain Monte Carlo scheme from a comparison of dogbone tensil e test and simulations. (b) A qualitative comparison of the calibration on the same dogbo ne geometry. 3.2. Calibration Using Global Load and Full-ﬁeld Strain Data The results of calibration for the second example where global load and the full-ﬁeld surface strain from DIC were used as the QoI are listed in T able 2. As previously noted, BI ﬁtrefers to the calibration that was performed with the library of simulations only and BI + ML ﬁt refers to the calibration where ML regression model was used to enrich the simulation library. The r esults for BI ﬁt are graphically shown in Figure 4. Overall, the elastic modulii ar e cali- brated with a higher conﬁdence compared to the other four parameters. T his is evident from the relatively narrow shape of the marginal posterior probab ility 14 distribution for EAandEMin Figure 4(a). The least-squares calibrated param- eters furnish results that provide the closest match with the ex perimental data. The second closest match is furnished by the MAP parameters from the B I ﬁt. This is evident in the MAPE values in the table and in the load-displac ement curves in Figure 4(b). The local shear strain ( εxy) prediction with the median ﬁtted parameters is compared with the DIC results at peak load in Figure 4(c). The qualitative comparison appears good. However, quantitatively the s trains match only modestly. For example, if MAPE were calculated only for the l oad quantities of interest, then it is 11% for the median ﬁtted parameters . This means that a larger error (MAPE) is contributed by strains than load. We discuss this aspect further in the Discussion section below. The results for BI + ML ﬁt are graphically shown in Figure 5. Both median and MAP estimates from the BI + ML approach provided a better ﬁt than the BI approach and a marginally better ﬁt than the least-squares approach. This is evident from the smaller MAPE in Table 2, narrower marginal poste rior probability distributions for the ﬁtted parameters in Figure 5(a) comp ared to the BI ﬁt, and a relatively close match between the simulated load-di splacement curves for the ﬁtted parameter and the experimental data in Figure 5(b ). The local shear strain prediction with the median ﬁtted parameters is com pared with the DIC results at peak load in Figure 5(c). The qualitative comparison app ears good. Fit description EaEmεtσUPSσLPSσCPSMAPE (GPa) (GPa) (MPa) (MPa) (MPa) (%) Least squares ﬁt 49.4 39.2 0.0374 341 204 434 17.9 Bi ﬁt (median) 56.1 35.7 0.0453 402 194 523 20.5 BI ﬁt (MAP) 55.9 35.6 0.0524 470 108 475 22.2 BI ﬁt (95% credible interval)40.3 21.9 0.0355 305 105 359 75.9 41.4 0.0545 495 294 692 BI+MLﬁt(median) 54.0 42.7 0.0377 355 204 407 17.3 BI + ML ﬁt (MAP) 54.4 44.9 0.0368 351 201 408 17.4 BI + ML ﬁt (95% credible interval)43.1 31.4 0.0351 328 191 379 66.3 49.5 0.0442 387 218 437 Table 2: Results of calibration for the diamond specimen usi ng load and full-ﬁeld strain data. The use of both strain and load QoI in this example, rather than using just the macro load as the QoI, resulted in the calibrated material parameters that furnished a more accurate simulation of the local strains. We performe d a BI + ML calibration using only the load QoI. This resulted in median mcal= {48.9 GPa, 31.2 GPa, 0.0447, 342 MPa, 194 MPa, 494 MPa }. These values are diﬀerent than the results listed in Table 2 for the median mcal. These material parameters resulted in a MAPE of 21.9%, which is larger than the MAPE from median BI + ML parameters listed in the table (17.3%). The MAPE calculate d for load values only was 3.71%, which is smaller than the MAPE for load from median BI + ML parameters listed in the table (5.3%). This means that th e use of only load QoI resulted in a more accurate simulation of the load respons e, 15 but a less accurate simulation of the local strain response. Figure 4: Results of BI calibration with global load and surf ace strains as the quantity of inter- est. (a) Marginal posterior probability distributions of t he six NiTi material model parameters sampled using the Markov-chain Monte Carlo scheme from a com parison of diamond tensile test, DIC strain ﬁeld data, and simulations. (b) A qualitati ve comparison of the calibration on the same diamond geometry. Load-displacement data from t he tensile test is shown. (c) Comparison of the local shear strain at peak displacement du ring the tensile test as measured using DIC and from a simulation with the median ﬁtted materia l properties. The results of a simple validation test performed on a planar dogbone spec - imen using the material parameters obtained from the BI + ML approach are shown in Figure 6. Simulated and experimental load-displacement cur ves match reasonably well. MAPE for median and MAP ﬁtted parameters is 12.2% and 12.1% respectively. The model calibration was performed using axial load and local shear strain data with the specimen in a mixed loading mode. Th e load- ing mode in this validation test was axial. Considering that the calibr ation was performed in the mixed loading mode vs. the validation with uniaxial l oading mode, this validation shows that the calibration is robust. The dogbone s peci- men used in this simple validation study is diﬀerent compared to th e dogbone specimen used in Section 3.1. This dogbone was manufactured from a planar Nitinol strip material, while the dogbone in Section 3.1 was manufactured from a Nitinol tubing material. Nitinol strip and tube materials exhibit di ﬀerent mechanical properties. 4. Discussion 4.1. Propagation of Material Parameter Uncertainty to Simulatio ns of Cyclic Loading The material parameter uncertainties obtained using the BI + ML cali- bration approach above can be propagated to any subsequent simulations of 16 Figure 5: Results of BI + ML calibration with global load and s urface strains as the quantity of interest. (a) Marginal posterior probability distribut ions of the six NiTi material model parameters sampled using the Markov-chain Monte Carlo sche me from a comparison of dia- mond tensile test, DIC strain ﬁeld data, and simulations. Th e simulation library was enriched using a machine learning surrogate model in this case. (b) A q ualitative comparison of the calibration on the same diamond geometry. Load-displaceme nt data from the tensile test is shown. (c) Comparison of the local shear strain at peak displ acement as measured using DIC and from a simulation with the median ﬁtted material propert ies. Figure 6: A simple validation of the BI + ML calibrated parame ters obtained from the diamond geometry. The validation simulation was performed on a planar dogbone simulation geometry. Theexperimentaldatawereobtainedfromtensile testofaplanardogbonespecimen fabricated from the identical material as that used to make t he diamond samples. superelastic deformation. We discuss the consequences of propagating the ma- terialparameteruncertainties. Oneofthemostcommonpurposesofsimu lations using the superelastic model is to assess the fatigue performance of a medical device fabricated from NiTi [29]. The simulations designed for this pur pose typ- 17 ically impose a mean displacement and a cyclic displacement amplit ude on the component as schematically shown in Figure 7(a). The distribution of th e local mean strain and strain amplitude in the device during a fatigue loading cycle is then simulated, resulting in a point cloud of the strains as shown in Figure 7(b). The critical mean strain and the critical strain amplitude in the poin t cloud are typically used as the fatigue indicator parameters for the NiTi devic e and a fatigue safety factor (FSF) is calculated from them. Any uncertainty i n the simulation inputs can naturally introduce an uncertainty in the FSF calculation. To demonstrate the utility of the BI calibration approach presented he re, we illustrate how the uncertainty in the superelastic NiTi material p arameters can propagatetoasimulationoffatiguedeformation. Wecreatedasimulationofth e diamond geometry shown in Figure 2(b) with an imposed mean displacement of 1.0 mm and a displacement amplitude of 0.4 mm. We created 200 instances of this simulation with material parameters sampled within the 95% cred ible interval for the BI + ML ﬁt described in Section 3.2. We extracted the strain tensor at each integration point in the model at the two extrema of the fat igue loading cycle marked by D and E in Figure 7(a). We calculated the mean strai n Emeanand the strain amplitude Eampat each integration point as, Emean=1 2(ED+EE),Eamp=1 2(ED−EE), (5) whereEDandEEis the strain tensor at an integration point at D and E re- spectively. For each simulation, we calculated the maximum princip al invariant of the two strain tensors above at each integration point. A sample scatter plot of the strain amplitude plotted against mean strain is shown in Figure 7(b ). We determined the critical strain amplitude and the critical mean s train at the point with the largest value of the maximum principal invariant of Eamp. The probability distribution of the critical mean strain from the 200 si m- ulations is shown in Figure 7(c) and the probability distribution of th e strain amplitude is shown in Figure 7(d). In both plots, the result obtained from the median calibrated material properties from the BI + ML approach is marked by a dashed line. The mean strain shows a range of approximately 0.01 and th e strain amplitude shows a range of approximately 0.007. FSF calculated from this will show a relatively large variation. Thus, we have demonstrat ed that the uncertainty information furnished by the BI-based calibration approac h can be propagated to fatigue simulations of NiTi and it can result in appreciable un - certainty in the fatigue indicator parameters. Similar to strains, if other fatigue indicator parameters for NiTi such as the phase transformation volume ampli - tude [30] are extracted from the simulations, then those values will be aﬀected by the material parameter uncertainty as well. We emphasize that this approach provides a quantiﬁcation of the material parameter uncertainty that may be propagated to any subsequent simulat ions of fatigue. However, this approach does not quantify the accuracy of the FSF cal- culated in this subsequent simulation. The accuracy of the FSF will be certainly dependent on the accuracy of the calibrated material parameters. Howeve r, it willalsobeinﬂuencedbytheabilityoftheinherentmaterialmode linsimulating 18 Figure 7: A demonstration of the propagation of material par ameter uncertainties to a subse- quent simulation. (a) Schematic load-displacement curve o f a fatigue-type loading character- ized by a mean displacement and a displacement amplitude. (b ) A representative scatter plot of the strain amplitude vs. mean strain at ﬁnite element mode l integration points calculated from the loading states D and E shown in (a). (c) Distribution of mean strain and (d) strain amplitude for fatigue loading simulations performed with t he material parameter probability distribution obtained using the BI + ML approach. the fatigue indicator parameters, the accuracy of the underlying ﬁnit e element model (e.g., geometry and mesh resolution), and any other assumptions mad e in the simulations. 4.2. Accuracy of Local Compressive Strain Prediction from the Cal ibrated Pa- rameters While using the BI approach has beneﬁts over the least-squares calib ration approach and using the diamond geometry for calibrating the NiTi superelas - ticity model has beneﬁts over the use of only simple tension test d ata, neither the BI approach nor the specimen geometry can overcome the limitations of the underlying constitutive model. In this section we discuss one key example of such limitations. One motivation behind using the full-ﬁeld sur face strain data was to obtain such calibrated material parameters that will furnish a more accurate simulated local strain distribution compared to the calibrat ion with just the global load data. We showed above that the BI + ML approach using strain and load QoI indeed furnished more accurate local strains in term s of the MAPE compared to when a similar calibration was performed using only the load QoI. However, it is worth comparing the BI + ML calibration results w ith the DIC data in detail. Thus, we compare the simulated local strain di stribution 19 from the optimum calibrated material parameters with the experiment al data obtained using DIC. To emphasize, these strains are for the diamond BI + ML calibration results when both strain and load QoI were used. A comparison of the simulated local shear strains (represented in ter ms of simulation QoI) with the DIC measurements (represented in terms of experi- mental QoI) in two key regions of the diamond sample at the peak load is shown in Figure 8. As described in Section 2.3.2, strains averaged over four key re- gions in the diamond geometry served as the QoIs along with the global load in the demonstration of BI + ML calibration approach. Two of those regions – marked 2 and4 in Figure 2(c) – exhibit compressive strains. In the ﬁgure, we have plotted the simulated strains in these two regions as a funct ion of the ratioσCPS/σUPS. The relative error in the strain prediction from the median or MAP material parameter ﬁt with respect to the experimental measure ment is up to 25%. While this is undesirable, it is unavoidable consideri ng that the model consistently underestimated the strains in some regions irr espective of the material parameters used as shown in Figure 8(b). The phenomenologic al constitutive model considered in this study approximates the NiT i deformation as isotropic, among other assumptions [16]. As a consequence of this, the ac- curacy of the simulated local compressive strains from the calibrated material properties is modest. In the same ﬁgure, it can be seen that the ratio σCPS/σUPSis 1.3 for the median ﬁt and 1.0 for MAP from the BI + ML approach with the 95% credible interval values in between. While the ratio is typically taken as 1.5 for NiTi in tube form, 1.0 to 1.3 is a reasonable number for NiTi strip material. Th e strip material has a texture with a strong ∝an}bracketle{t110∝an}bracketri}htcomponent [31]. A simple calculation using the crystallographic theory of martensite [32] and mech anics reveals that σCPS/σUPS= 1.0 for [110] orientation and the ratio increases for orientationstowards[111], ultimatelyreaching1.5(SeeSupplemen taryDatafor details of the calculation). Thus, while the local compressive strai n predictions with the ﬁtted material parameters are lacking due to the model limi tations, theσCPS/σUPSratio is reasonably calibrated. The MAP estimate of 1.0 for this ratio is also consistent with the observations in the literature for [ 110] oriented single crystals [22]. 4.3. Advantages and Challenges of the BI Calibration Approach A clear beneﬁt of the BI approach to calibrating constitutive models i s the information about uncertainty in the ﬁt obtained from them. This materi al parameter uncertainty, together with other uncertainties can be prop agated to subsequent simulations as we demonstrated. The beneﬁt of using an ML- based surrogate model is that it can improve the parameter ﬁt by enriching t he sim- ulation library used in the calculation of likelihood function. The ap proach presented here is quite general and can be implemented for any model for su- perelastic deformation. Also, this approach can incorporate essentially an y QoI in the calibration process as long as it can be extracted from the simulat ions built with that constitutive model. In particular, the use of strai n QoI resulted in a more accurate simulation of the local strains compared to using only t he 20 Figure 8: Accuracy of the compressive local strain predicti on using the calibrated material properties. (a) Comparison of simulated strain QoI with exp erimental (DIC) strain QoI in region2 in Figure 2(c) at peak load shown in Figure 2(b) by point B. (b ) Comparison of simulated strain QoI with experimental (DIC) strain QoI in r egion4 in Figure 2(c) at peak load. The simulated strains are lower in magnitude than the D IC measurement which is shown by a black horizontal line in both plots. The QoIs are ca lculated using the shear strain component. load as the experimental comparator. The diamond geometry furnishes both tensile and compressive response in diﬀerent regions and allows the calibration of both tensile and compressive model parameters from a single test. A key challenge associated with this approach compared to the least-squ are or trial-and-error calibration is the relatively large upfront eﬀort req uired in setting up the calibration scheme. It requires the construction of a simulation library for a speciﬁc test geometry and test method speciﬁcation. How ever, once such a library is established, it can be used for any subsequent cali bration using the same test setup. An informed choice between this sophisticated calibration method and alternatives can be made by setting concrete expectations about the credibility of simulations for a speciﬁc context of use. Framewor ks such as the ASME V&V 40 standard can be used for this purpose [33]. For example, if the model risk is high for a particular context of use, then invest ing in the BI calibration scheme could be desirable to ensure the inﬂuence of m aterial parametersandtheiruncertaintyisrigorouslyconsidered. However, ifthemodel risk is low, then a simpler calibration approach may be used to obtain t he nominal material properties. Implementation and execution of this calibration method requires ce rtain subjective inputs on the part of the analyst. For example, prior distr ibution of the material parameters is a subjective choice. We demonstrated th at the calibrated material parameters from the posterior distribution can be e stimated using point estimates such as median and MAP. However, it is a subject ive choice to select a particular point estimate. The weights used in cal culating the likelihood function are also a subjective choice. However, th ese subjective choices present an opportunity to incorporate the specialist’s know ledge about the constitutive response in the calibration scheme. 21 Since the likelihood function is calculated from a comparison of the s imu- lated and the experimentally observed response of the QoIs, the accur acy of calibration may be impacted by the limitations of the test setup and th e lim- itations of the underlying model itself. For example, an incorrect D IC setup can furnish an inaccurate surface strain distribution, which will i mpact the cal- ibration. In Section 4.2 we reported the modest accuracy of the local st rains simulated using the calibrated parameters due to the limitations of t he model itself. We also attempted to use the full strain tensor in four region s (1 -4 in Figure 2(c)) as the QoI in the diamond calibration example. However, th e results were not superior to the case when the shear strain component was used as the quantity of interest. We found that the ﬁnite element model u sed in this study was able to capture the shear strains more accurately compared to t he normal strain components (See supplementary Data for additional details ). Ad- ditionally, the SVM surrogate model accuracy was lower for the normal strai n components. These factors plausibly lead to the strain tensor-based c alibra- tion approach not yielding superior results to the approach using only t he shear strain component. Thus, ensuring that the test methods furnish r obust data, appropriate QoIs are selected to perform the calibration, and the under lying forward model is accurate are the responsibilities of the specialist to ensure that reliable calibrated material parameters are obtained and the simulation r esults from such parameters are accurate. There are several opportunities to further reﬁne this calibration ap proach and the communication of calibration results. There is signiﬁcant prior work on incorporating individual sources of uncertainty in the Bayesian cal ibration procedure [5, 34]. This work can be extended to individually assess t he impact of various sources of uncertainty on the calibrated parameters. We exten sively documented the eﬀect of model error on the accuracy of calibration. Vari ous eﬀorts have suggested methods to account for and reduce the impact of mo del error or uncertainty on the calibration procedure [4, 35]. We presented the uncertainty in the calibrated material parameters in terms of the cre dibility interval. The credibility intervals may be reported in various wa ys including the highest posterior density interval (HPI) or equal-tailed interv al [8]. One of such representations that is the most informative may be chosen when reporting the uncertainty. 5. Summary and Conclusions We presented the implementation of a Bayesian Inference material p roperty calibration approach for a constitutive model to simulate the superel astic de- formation of NiTi shape memory alloy. We made three contributions as part of this approach: 1. We presented a diamond standard specimen geometry that can be used to calibrate both tensile and compressive material parameters. This i s signiﬁcantconsideringthattheconstitutiveresponseofNiTiisasym metric in tension vs. compression. 22 2. We demonstrated a method to incorporate full-ﬁeld surface strain d ata obtained using digital image correlation in the calibration process. Thi s is signiﬁcant because full-ﬁeld strain data furnishes a large numbe r of comparators in the calibration process and can contribute to increasing the accuracy of the calibration. Incorporation of surface strain data also resulted in model parameters that furnish more accurate local strain dis- tributions and a more accurate determination of the fatigue safety factor from the simulations. 3. Wedemonstratedthatarelativelysimplemachinelearningsurrogate model canbeusedtoenrichthe simulationlibraryusedinthecalibration. Thisis signiﬁcant because generating a simulation library using a nonlinear c on- stitutive law such as that used in this work is challenging. The surr ogate modelcanenableperforminghigh-qualitycalibrationinacomputational ly eﬃcient manner. The key advantage of the Bayesian Inference approach is that it furnishe s un- certainty in the calibrated material parameters and that uncertainty c an be propagated to subsequent simulations. While we applied the calibrati on scheme to the Auricchio and Taylor [16] model for superelastic response impl emented in Abaqus ﬁnite element modeling framework, this method is versatil e and can be appliedtoothermodelsforthedeformationofNiTiSMA.Theultimateacc uracy of simulations conducted with the calibrated material parameters, how ever, de- pends on the ability of the underlying constitutive model to accur ately capture the physics of the material response. 6. Data Availability The raw data required to reproduce these ﬁndings cannot be shared at this time due to technical or time limitations. The processed data requ ired to re- produce these ﬁndings are available to download from https://github.com/ confluentmedical/nitinol-bayes-cal . Annotated computer code to per- form this calibration is available at https://github.com/confluentmedical/ nitinol-bayes-cal . 7. CRediT Author Statement Harshad M. Paranjape: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - Original draft. Kenneth I. Aycock : Methodol- ogy, Writing - Original draft. Jason D. Weaver, Brent A. Craven: Metho dology. Craig Bonsignore, Thomas W. Duerig: Resources. All authors: Writing - Re - view & Editing. 8. Acknowledgement HMP would like to thank Conﬂuent Medical Technologies, Inc. for testin g and computational resources. The ﬁndings and conclusions in this articl e have 23 not been formally disseminated by the U.S. FDA and should not be constru ed to represent any agency determination or policy. The mention of commercial prod- ucts, their sources, or their use in connection with material report ed herein is not to be construed as either an actual or implied endorsement of such pr oducts by the Department of Health and Human Services. 9.