Comparison of Bayesian methods on parameter identification for a viscoplastic model with damage

Adeli, Ehsan; Rosić, Bojana; Matthies, Hermann G.; Reinstädler, Sven; Dinkler, Dieter

doi:10.1016/j.probengmech.2020.103083

Cited by 11 publications

(10 citation statements)

References 56 publications

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“…To enhance its efficiency, several techniques have been recently developed, such as delayed rejection adaptive Metropolis (DRAM) [57], differential evolution adaptive Metropolis (DREAM) [58], and ensemble Kalman filter with MCMC (EnKF-MCMC) [59]. The reader is referred to [56,[60][61][62][63] for the application of Bayesian inversion in applied sciences.…”

Section: Bayesian Inversion As a Backward Modelmentioning

confidence: 99%

“…2.3.3). The local evolution of the plasticity variables {ε p , α} according to the evolution equation (61), which represents an incremental, primal version of equations (54) and (55), is then a necessary condition of the minimization principle (20). For the present model, the yield function (52) takes the form…”

Section: Local Plasticity With ã C Based Fracture Criteria: Model 2 (M 2 )mentioning

confidence: 99%

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Bayesian inversion for unified ductile phase-field fracture

et al. 2021

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The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the $$\hat{R}{-}convergence$$ R ^ - c o n v e r g e n c e tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

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Section: Bayesian Inversion As a Backward Modelmentioning

confidence: 99%

Section: Local Plasticity With ã C Based Fracture Criteria: Model 2 (M 2 )mentioning

confidence: 99%

Bayesian inversion for unified ductile phase-field fracture

et al. 2021

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show abstract

“…To enhance its efficiency, several techniques have been recently developed, such as delayed rejection adaptive Metropolis (DRAM) [52], differential evolution adaptive Metropolis (DREAM) [53], and ensemble Kalman filter with MCMC (EnKF-MCMC) [54]. The reader is referred to [51,55,56,57,58] for the application of Bayesian inversion in applied sciences.…”

Section: Bayesian Inversion As a Backward Modelmentioning

confidence: 99%

“…To derive the above governing equations from the incremental minimization problem (20), we make use of equations ( 42) and (56), such that the functional derivative of (19) with respect to {ε p , α} can be written as…”

Section: Plasticitymentioning

confidence: 99%

Bayesian inversion for unified ductile phase-field fracture

Noii¹,

Khodadadian²,

Ulloa³

et al. 2021

Preprint

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The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity-and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis-Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the R − convergence tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

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“…Ctw=kt Q=@ pOt VwQ '=yxR=U O=tDa= C}r@=k p}rLD w |vt}= |Dq=tDL= p}rLD QO Qov=}@ R "CU= xOW xH=wt u= QoWywSB u}@ QO |O=}R p=@kDU= =@ '1 |x]@= Q j@] "CU= Q=@ Q=F; =} VvD |xOv}=tv S w sDU}U l} =} 'w[a 'xirwt l} Ctw=kt g(R; S) = R S (1) wO x@ g(R; S) = 0 |oDN}Uo `@=D \UwD (R; s) 'xrUt |wHwCUH |=[i w ut}= Cr=L QO xR=U |=yOQmrta "OwW|t s}UkD ut}=Q}e w ut}= |x}L=v :CU= xOW x=Q= 3 w 2 |=yx]@= Q QO ? }DQD x@ ut}=Q}e g(R; S) 0 (2) g(R; S) 0 (3) xOW XNWt ut}=Q}e w ut}= |L=wv w |iO=YD |=yQ}eDt |=[i '2 pmW QO p=Ut QO ?re= xm g(R; S) |OQmrta `@=D =} |oDN}Uo |OL `@=D u}tND "CU= |vD@ OU |}xRQr |Dq=tDL= p}rLD |xar=]t QO '2014 u= Q=mty w 2 uJ RmQtDt sQH R= 'uRNt |R=UpOt |=H x@ w ^=Lr =tO Q=@ QF= 'u}J nv=}=O |Uwk |xv}W}@ VvD `@=wD "CU= xOW xO=iDU= pO=at sQH CQwYx@ OU CUOq=@ QO `k=w "Ov=xOW xDiQo Q_v QO |@=QN |OL Cr=L `@=wD u=wva x@ R}v |Q=Wi w |WWm Ctw=kt |= Q@ w 10 'uw}U=Ovwi w xvO@ uD@ |v=UWm pwOt |= Q@ C= Q}}eD ?} Q[ Qy |= Q@ |vt}= XN=W "CU= xOW xDiQo Q_v QO 30 [7] "CU= |Q=Wi |Q=PoQ=@ xOWv ^=Lr l}D=DU=wQO}y ?…”

mentioning

confidence: 99%