Phase-field modeling of fracture has gained popularity within the last decade due to the flexibility of the related computational framework in simulating three-dimensional arbitrarily complicated fracture processes. However, the numerical predictions are greatly affected by the presence of uncertainties in the mechanical properties of the material originating from unresolved heterogeneities and the use of noisy experimental data. The objective of this work is to apply the Bayesian approach to estimate bulk and shear moduli, tensile strength and fracture toughness of the phase-field model, thus improving accuracy of the simulations with the help of experimental data. Conventional approaches for estimating the Bayesian posterior probability density function adopt sampling schemes, which often require a large amount of model estimations to achieve the desired convergence, thus resulting in a high computational cost. In order to alleviate this problem, we employ a more efficient approach called sampling-free linear Bayesian update, which relies on the evaluation of the conditional expectation of parameters given experimental data. We identify the mechanical properties of cement mortar by conditioning on the experimental data of the three-point bending test (observations) in an online and offline manner. In the online approach the parameter values are sequentially updated on the fly as the new experimental information comes in. In contrast, the offline approach is used only when the whole history of experimental data is provided once the experiment is performed. Both versions of estimation are discussed and compared by validating the phase-field fracture model on an unused set of experimental data. 由于相关计算框架在模拟三维任意复杂断裂过程方面的灵活性，断裂相场建模在过去十年中得到了普及。然而，由于未解决的异质性和噪声实验数据的使用，材料机械性能存在不确定性，数值预测受到很大影响。这项工作的目的是应用贝叶斯方法来估计相场模型的体积模量和剪切模量、拉伸强度和断裂韧性，从而借助实验数据提高模拟的准确性。传统的贝叶斯后验概率密度函数估计方法采用采样方案，通常需要大量的模型估计才能达到期望的收敛性，从而导致计算成本较高。为了缓解这个问题，我们采用了一种更有效的方法，称为无采样线性贝叶斯更新，它依赖于对给定实验数据的参数的条件期望的评估。我们通过在线和离线方式对三点弯曲试验（观察）的实验数据进行调节来确定水泥砂浆的力学性能。在在线方法中，参数值随着新的实验信息的进入而连续更新。相比之下，离线方法仅当在执行实验后提供实验数据的整个历史记录时才使用。通过在一组未使用的实验数据上验证相场断裂模型，讨论和比较了两种版本的估计。
The prediction of fracture is of main concern in many fields of engineering and the goal of several computational methods relying on different modeling frameworks. In the past two decades, a great amount of efforts were made to improve the accuracy of the numerical predictions mainly by (i) developing a more reliable fracture model and computational framework and (ii) identification of mechanical parameters with the help of experimental data. 断裂预测是许多工程领域的主要关注点，也是多种依赖不同建模框架的计算方法的目标。在过去的二十年中，人们为提高数值预测的准确性做出了大量努力，主要通过（i）开发更可靠的断裂模型和计算框架以及（ii）借助实验数据识别力学参数。
Computational approaches for simulating the fracture behavior of brittle materials are generally classified into three categories: discrete crack models [1], lattice models [2,3] and damage models [4]. In the discrete crack models, explicit modeling of cracks requires the mesh update by introducing 模拟脆性材料断裂行为的计算方法通常分为三类：离散裂纹模型[1]、晶格模型[2,3]和损伤模型[4]。在离散裂纹模型中，裂纹的显式建模需要通过引入来更新网格
new boundaries as the crack propagates. The development of the extended finite element method (XFEM), enriching the standard shape functions with discontinuous fields, can avoid remeshing [5]. However, the XFEM is very onerous to implement in three dimensions. In the cohesive zone model (CZM), crack paths are assumed to be known a priori and the insertion of cohesive elements between each pair of continuum elements in a mesh for unknown crack paths may lead to over-estimated crack areas [6]. An alternative is the so-called lattice model, in which the simulation results strongly depend on the chosen fracture criterion and element type [2,3,7]. In the local/gradient damage models, the stress-strain relation is hypothesized with the information coming from experiments [4]. Comprehensive summaries on this topic can be found in [8] and [9], and the references therein. 随着裂纹扩展出现新的边界。扩展有限元方法（XFEM）的发展丰富了具有不连续场的标准形函数，可以避免重新划分网格[5]。然而，XFEM 在三个维度上实施起来非常困难。在粘性区域模型 (CZM) 中，假设裂纹路径是先验已知的，并且在未知裂纹路径的网格中每对连续体元素之间插入粘性元素可能会导致高估裂纹面积 [6]。另一种方法是所谓的晶格模型，其中模拟结果很大程度上取决于所选的断裂准则和单元类型 [2,3,7]。在局部/梯度损伤模型中，应力-应变关系是根据来自实验的信息进行假设的[4]。关于这个主题的全面总结可以在 [8] 和 [9] 以及其中的参考文献中找到。
Phase-field modeling of fracture, stemming from Francfort and Marigo’s variational approach [10] is a very elegant and powerful framework to predict cracking phenomena. In this setting fracture is described by using a continuous field variable (order parameter) which provides a smooth transition between intact and fully broken material phases, and thus, fully avoids modeling cracks as discontinuities, see e.g. [11-15]. Purely based on energy minimization, the phasefield framework allows simulations of three-dimensional complicated fracture processes, including crack initiation, propagation, merging and branching in a unified framework without the need for ad-hoc criteria and remeshing. The model shares several features with gradient damage models [16]. 断裂相场建模源于 Francfort 和 Marigo 的变分方法 [10]，是预测裂纹现象的一个非常优雅且强大的框架。在此设置中，断裂是通过使用连续场变量（阶次参数）来描述的，该变量提供完整和完全断裂材料相之间的平滑过渡，从而完全避免将裂纹建模为不连续性，请参见例如[11-15]。纯粹基于能量最小化，相场框架允许在统一框架中模拟三维复杂断裂过程，包括裂纹萌生、扩展、合并和分支，而不需要临时标准和重新划分网格。该模型与梯度损伤模型有几个共同特征[16]。
In this work, we are concerned with cement mortar, which is known to be brittle or quasi-brittle. In brittle materials, the force resisted across a crack is zero, whereas quasibrittle materials feature a softening behavior with a cohesive response. For both materials, no apparent plastic deformation occurs before fracture. Wu et al. [17] provided a qualitative and quantitative comparison between mixed-mode fracture test results in cement mortar and numerical simulations through the phase-field approach, and a good agreement was found. 在这项工作中，我们关注的是水泥砂浆，众所周知，水泥砂浆是脆性或准脆性的。在脆性材料中，裂纹上的抵抗力为零，而准脆性材料具有具有内聚响应的软化行为。对于这两种材料，断裂前不会发生明显的塑性变形。吴等人。 [17]通过相场方法对水泥砂浆混合模式断裂试验结果与数值模拟结果进行了定性和定量比较，结果吻合良好。
The deterministic phase-field model, however, is not very suitable for the prediction of cracking in materials with unresolved heterogeneities. An example is cement mortar modeled as an homogeneous material but in fact exhibiting a heterogeneous and porous structure at a lower scale. The reason is that the parameterization of the model strongly depends on material characteristics which are not fully known. The choice of the parameter values certainly affects the accuracy of numerical simulations and thereby the estimation of the phase-field parameters given experimental data is required. One widely used and simple way to tune the parameters is to minimize the measured distance between the experimental data and their numerically predicted counterpart. However, 然而，确定性相场模型不太适合预测具有未解决的异质性的材料的裂纹。一个例子是水泥砂浆被建模为均质材料，但实际上在较低尺度上表现出异质和多孔结构。原因是模型的参数化强烈依赖于不完全已知的材料特性。参数值的选择肯定会影响数值模拟的准确性，因此需要根据实验数据估计相场参数。一种广泛使用且简单的参数调整方法是最小化实验数据与其数值预测对应数据之间的测量距离。然而，
the resulting optimization problem is often ill-posed and requires regularization. 由此产生的优化问题通常是不适定的并且需要正则化。
The generalization of the optimization methods to the Bayesian approach offers a natural way of combining prior information with experimental data for solving inverse problems within a coherent mathematical framework [18,19]. In this framework the knowledge about the parameters to be identified is modelled probabilistically, thus a priori reflecting an expert’s personal judgment and/or current state of knowledge about material characteristics. The information gained from the measurement data is used to reduce the uncertainty in the prior description via Bayes’s rule [19]. The posterior is thus the probabilistic prior conditioned on the measurement data. Such an approach is then more suitable than deterministic optimization approaches as if our measurement data do not contain information about the parameter set then Bayes rule will output prior description, whereas deterministic approaches will possibly become unstable and output non-physical values. Therefore, deterministic approaches usually use some type of regularization which further defines the parameter values. 将优化方法推广到贝叶斯方法提供了一种将先验信息与实验数据相结合的自然方法，用于在连贯的数学框架内解决反演问题[18,19]。在该框架中，关于要识别的参数的知识被概率建模，因此先验地反映了专家的个人判断和/或关于材料特性的知识的当前状态。从测量数据中获得的信息用于通过贝叶斯规则[19]减少先前描述中的不确定性。因此，后验是基于测量数据的概率先验。这种方法比确定性优化方法更合适，因为如果我们的测量数据不包含有关参数集的信息，那么贝叶斯规则将输出先前的描述，而确定性方法可能会变得不稳定并输出非物理值。因此，确定性方法通常使用某种类型的正则化来进一步定义参数值。
The evaluation of Bayes’s posterior is always challenging. The posterior probability densities can be computed via the Markov chain Monte Carlo (MCMC) integration. However, the method is slowly converging and may lead to high computational costs especially when nonlinear evolution models such as the phase-field model are considered [20]. With the intention of accelerating the MCMC method, the transitional MCMC [21] is often used. The idea is to obtain the samples from series of simpler intermediate PDFs that converge to the target posterior density and are faster to evaluate. Further acceleration of MCMC-like algorithms can be achieved by constructing the proxy-(meta-/surrogate) models via e.g. a polynomial chaos expansion (PCE) and/or a Karhunen-Loève expansion (KLE) [22-24], both of which are cheap to evaluate. 贝叶斯后验的评估始终具有挑战性。后验概率密度可以通过马尔可夫链蒙特卡罗（MCMC）积分来计算。然而，该方法收敛缓慢，可能导致较高的计算成本，尤其是在考虑相场模型等非线性演化模型时[20]。为了加速 MCMC 方法，经常使用过渡 MCMC [21]。这个想法是从一系列更简单的中间 PDF 中获取样本，这些样本收敛到目标后验密度并且评估速度更快。类似 MCMC 算法的进一步加速可以通过构建代理（元/代理）模型来实现，例如通过多项式混沌展开（PCE）和/或 Karhunen-Loève 展开（KLE）[22-24]评估起来很便宜。
The approaches mentioned above result in the samples from the posterior density as a final outcome. However, a more efficient approach is to focus on posterior estimates that are of engineering importance, such as the conditional mean value or maximum a posteriori point (MAP) in a case of unimodal posterior. In this manner, the posterior random variable representing our knowledge about the parameter mean value can be evaluated via algorithms that are based on approximations of Bayes’s rule [19]. These algorithms give the posterior random variable (RV) as a function of the prior RV and the measurement. The simplest approximation is to try to get the conditional mean correct, as this is the most significant descriptor of the posterior. Such algorithms result from an orthogonal decomposition in the space of random variables [25], which upon restriction to linear maps gives the Gauss-Markov-Kalman filter. Numerical realisations of such maps can be computed again via sampling resulting in the so-called 上述方法产生后验密度样本作为最终结果。然而，更有效的方法是关注具有工程重要性的后验估计，例如单峰后验情况下的条件平均值或最大后验点（MAP）。通过这种方式，代表我们对参数平均值的了解的后验随机变量可以通过基于贝叶斯规则近似的算法来评估[19]。这些算法给出后验随机变量 (RV) 作为先验 RV 和测量值的函数。最简单的近似是尝试获得正确的条件均值，因为这是后验最重要的描述符。此类算法源自随机变量空间中的正交分解[25]，在限制线性映射的情况下给出高斯-马尔可夫-卡尔曼滤波器。可以通过采样再次计算此类映射的数值实现，从而产生所谓的
ensemble Kalman filter [26], or the computations can be performed in the framework of a polynomial chaos expansion (PCE) [25]. The latter one allows a direct algebraic way of computing the posterior estimate without any sampling at the update level. Therefore, the computational cost is greatly reduced, while the accuracy of the posterior estimate can be guaranteed as discussed in papers with a broad range of applications, e.g. diffusion [19], elasto-plasticity [19] and damage [27]. 集成卡尔曼滤波器[26]，或者可以在多项式混沌展开（PCE）[25]的框架中执行计算。后一种方法允许采用直接代数方式计算后验估计，而无需在更新级别进行任何采样。因此，计算成本大大降低，同时可以保证后验估计的准确性，如具有广泛应用的论文中所讨论的那样，例如扩散[19]、弹塑性[19]和损伤[27]。
Applying the Bayesian approach for parameter identification of the mechanical models can be traced back to the 1970s [28], when elastic parameters describing linear mechanical behavior were estimated. Later on, this framework has been employed for various linear problem applications, e.g. mechanical response of composite [29], dynamic response [30], as well as thermal and diffusion problems [19,31]. The extension to nonlinear models appeared later on. Rappel et al. [32] adopted the Bayesian approach to identify elastoplastic material parameters of one-dimensional single spring based on uniaxial tensile results, in which the probability density function (PDF) of the posterior was calculated by MCMC with an adaptive Metropolis algorithm. They accounted for the statistical errors which polluted the measurements not only in stress but also in strain. Rosić et al. [19] have applied the sampling-free linear Bayesian update via PCE to estimate the parameters in the elastoplastic model. The method in different variants was thoroughly discussed on a nonlinear thermal conduction example in [31]. Huang et al. [33] have utilized two Gibbs sampling algorithms for Bayesian system identification based on incomplete model data with application to structural damage assessment. The equation-error precision parameter was marginalized over analytically, such that the posterior sample variances were reduced. 应用贝叶斯方法进行机械模型参数识别可以追溯到 20 世纪 70 年代 [28]，当时估计了描述线性机械行为的弹性参数。后来，该框架已用于各种线性问题应用，例如复合材料的机械响应[29]、动态响应[30]以及热和扩散问题[19,31]。后来出现了非线性模型的扩展。拉佩尔等人。 [32]采用贝叶斯方法基于单轴拉伸结果识别一维单弹簧弹塑性材料参数，其中后验概率密度函数（PDF）由MCMC和自适应Metropolis算法计算。他们解释了统计误差，这些误差不仅影响了应力测量，还影响了应变测量。罗西奇等人。 [19]通过PCE应用免采样线性贝叶斯更新来估计弹塑性模型中的参数。在[31]中的非线性热传导示例中彻底讨论了不同变体的方法。黄等人。 [33]利用两种吉布斯采样算法进行基于不完整模型数据的贝叶斯系统识别，并将其应用于结构损伤评估。方程误差精度参数在分析上被边缘化，从而减少了后验样本方差。
The objective of this work is to identify mechanical parameters of cement mortar for phase-field modeling of fracture by using a sampling-free linear Bayesian approach. The measurements which update the prior probability description to a posterior one come from three-point bending tests. However, this paper does not represent just simple combination of nonlinear mechanics model (phase-field model) and sample-free linear Bayesian approach. As the identification procedure is probabilistic and linear, the conditional estimate of the phase-field parameter set will not be optimal due to the nonlinear measurement-parameter relationship. Hence, we develop online sequential algorithm in which the updating is performed step-wise (one measurement at the time) such that the measurement-parameter relationship can be observed as close to linear as possible. This is further compared to the offline approach in which complete measurement history is taken into account for the parameter estimation. 这项工作的目的是通过使用免采样线性贝叶斯方法来确定水泥砂浆的力学参数，以进行裂缝相场建模。将先验概率描述更新为后验概率描述的测量来自三点弯曲测试。然而，本文并不代表非线性力学模型（相场模型）和无样本线性贝叶斯方法的简单组合。由于识别过程是概率性和线性的，由于非线性测量参数关系，相场参数集的条件估计将不是最优的。因此，我们开发了在线顺序算法，其中更新是逐步执行的（一次测量一次），以便可以观察到尽可能接近线性的测量参数关系。这进一步与离线方法进行比较，在离线方法中，参数估计考虑了完整的测量历史。
The paper is organised as follows. Section 2 introduces the basics on phase-field modeling of brittle fracture. In Sect. 3, the sampling-free linear Bayesian update of polynomial 本文的结构如下。第 2 节介绍了脆性断裂相场建模的基础知识。昆虫。 3、多项式的免采样线性贝叶斯更新
chaos representations is described. Section 4 is concerned with online and offline procedures for updating. In Sect. 5 the experimental setup of the three-point bending test is presented, and the updating process of the unknown material properties is described. The last section concludes the work with a discussion of the results and possible future work. 描述了混沌表示。第 4 节涉及在线和离线更新程序。昆虫。图5给出了三点弯曲试验的实验装置，并描述了未知材料属性的更新过程。最后一部分总结了工作，讨论了结果和未来可能的工作。
2 Phase-field modeling of brittle fracture 2 脆性断裂相场模拟
The phase-field approach used in this paper originates from the regularization of the variational formulation of brittle fracture by Francfort and Marigo [10]. For brittle materials, the fracture problem is formulated as the minimization problem of the energy functional 本文使用的相场方法源于 Francfort 和 Marigo [10] 对脆性断裂变分公式的正则化。对于脆性材料，断裂问题被表述为能量泛函的最小化问题 E(u,Gamma)=int_(Omega\\Gamma)psi_(el,0)(epsi(u))dx+G_(c)int_(Gamma)dGamma\mathcal{E}(\mathbf{u}, \Gamma)=\int_{\Omega \backslash \Gamma} \psi_{\mathrm{el}, 0}(\boldsymbol{\varepsilon}(\mathbf{u})) \mathrm{d} \mathbf{x}+\mathrm{G}_{\mathrm{c}} \int_{\Gamma} \mathrm{d} \Gamma,
in which ox subR^(n),n=2,3\otimes \subset \mathbb{R}^{n}, n=2,3, is an open bounded domain representing an elastic nn-dimensional body, Gamma sub ox\Gamma \subset \otimes is the crack set, psi_(el,0)(epsi)\psi_{\mathrm{el}, 0}(\boldsymbol{\varepsilon}) is the elastic strain energy density, G_(c)G_{c} is the fracture toughness (or critical energy release rate), and epsi=(1)/(2)(gradu+gradu^(T))\varepsilon=\frac{1}{2}\left(\nabla \mathbf{u}+\nabla \mathbf{u}^{T}\right) is the infinitesimal strain tensor with u\mathbf{u} as the displacement field. Introducing regularization as in Bourdin et al. [11], Eq. (2.1) can be recast as 其中 ox subR^(n),n=2,3\otimes \subset \mathbb{R}^{n}, n=2,3 ，是一个开有界域，表示弹性 nn 立体体， Gamma sub ox\Gamma \subset \otimes 是裂纹集， psi_(el,0)(epsi)\psi_{\mathrm{el}, 0}(\boldsymbol{\varepsilon}) 是弹性应变能密度， G_(c)G_{c} 是断裂韧性（或临界能量释放率），并且 epsi=(1)/(2)(gradu+gradu^(T))\varepsilon=\frac{1}{2}\left(\nabla \mathbf{u}+\nabla \mathbf{u}^{T}\right) 是无穷小应变张量 u\mathbf{u} 作为位移场。引入正则化，如 Bourdin 等人。 [11]，等式。 (2.1) 可以改写为 E_(l)(u,s)=int_(Omega)psi_(el)(epsi(u),s)dx+G_(c)int_(Omega)[(1)/(4ℓ)(1-s)^(2)+ℓ|grad s|^(2)]dx\mathcal{E}_{l}(\mathbf{u}, s)=\int_{\Omega} \psi_{\mathrm{el}}(\boldsymbol{\varepsilon}(\mathbf{u}), s) \mathrm{d} \mathbf{x}+\mathrm{G}_{\mathrm{c}} \int_{\Omega}\left[\frac{1}{4 \ell}(1-s)^{2}+\ell|\nabla s|^{2}\right] \mathrm{d} \mathbf{x},
in which ss is the phase-field parameter describing the state of the material. This parameter varies smoothly between 1 (intact material) and 0 (completely broken material), 0 <= s <=0 \leq s \leq 1. Here, ℓ\ell is a length scale parameter characterizing the width of the diffusive approximation of a discrete crack, i.e. the width of the transition zone between completely broken and intact phases. The elastic strain energy density affected by damage, psi_(el)\psi_{\mathrm{el}}, is given by 其中 ss 是描述材料状态的相场参数。该参数在 1（完整的材料）和 0（完全破碎的材料）之间平稳变化， 0 <= s <=0 \leq s \leq 1. 在这里， ℓ\ell 是表征离散裂纹的扩散近似宽度的长度尺度参数，即完全破裂和完整相之间的过渡区的宽度。受损伤影响的弹性应变能密度， psi_(el)\psi_{\mathrm{el}} ，由下式给出 psi_(el)(epsi,s)=g(s)psi_(el)^(+)(epsi)+psi_(el)^(-)(epsi)\psi_{\mathrm{el}}(\boldsymbol{\varepsilon}, s)=g(s) \psi_{\mathrm{el}}^{+}(\boldsymbol{\varepsilon})+\psi_{\mathrm{el}}^{-}(\boldsymbol{\varepsilon}),
in which the split into positive and negative parts is used to differentiate the fracture behavior in tension and compression, as well as to prevent the interpenetration of the crack faces under compression. Therefore, normal contact between the crack faces is automatically dealt with. 其中正负部分的分割用于区分拉伸和压缩下的断裂行为，并防止压缩下裂纹面的相互渗透。因此，会自动处理裂纹面之间的法向接触。
In this work we adopt the volumetric-deviatoric split of the density in Eq. (2.3) by Amor et al. [12]: 在这项工作中，我们采用方程中密度的体积偏态分裂。 （2.3）阿莫尔等人。 [12]： psi_(el)^(+):=(1)/(2)K(:tr(epsi):)_(+)^(2)+G(epsi_(dev):epsi_(dev))\psi_{\mathrm{el}}^{+}:=\frac{1}{2} K\langle\operatorname{tr}(\boldsymbol{\varepsilon})\rangle_{+}^{2}+\mathrm{G}\left(\boldsymbol{\varepsilon}_{\mathrm{dev}}: \boldsymbol{\varepsilon}_{\mathrm{dev}}\right), psi_(el)^(-):=(1)/(2)K(:tr(epsi):)_(-)^(2)\psi_{\mathrm{el}}^{-}:=\frac{1}{2} K\langle\operatorname{tr}(\boldsymbol{\varepsilon})\rangle_{-}^{2}