A Bayesian estimation method for variational phase-field fracture problems

We consider a smooth, open and bounded domain \(\Omega \subset \mathbb {R}^\delta ~(\text {here}~ \delta =2)\). In this computational domain, a lower dimensional fracture can be indicated by \(\mathcal {C}\subset \mathbb {R}^{\delta -1}\). In the following, Dirichlet boundaries conditions indicated as \(\partial \Omega _D := \partial \Omega \), and Neumann boundaries conditions are given on \(\partial _N \Omega := \Gamma _N \cup \partial \mathcal {C}\), where \(\Gamma _N\) is the outer boundary of \(\Omega \) and \(\partial \mathcal {C}\) is the crack boundary. The geometric setup including notations is illustrated in Fig. 1a. The surface fracture \(\mathcal {C}\) is estimated in \(\Omega _F\subset \Omega \subset \mathbb {R}^\delta \). A region without any fracture (i.e., an intact region) is indicated by \(\Omega _R:=\Omega \backslash \Omega _F\subset \Omega \subset \mathbb {R}^\delta \) such that \(\Omega _R\cup \Omega _F=\Omega \) and \(\Omega _R\cap \Omega _F=\varnothing \).

The variational phase-field formulation is a thermodynamically consistent framework to compute the fracture process. Due to the presence of the crack surface, we formulate the fracture problem as a two-field problem including the displacement field \(\textit{\textbf{u}}(\textit{\textbf{x}}):\Omega \rightarrow \mathbb {R}^\delta \) and the crack phase-field \(d(\textit{\textbf{x}}):\Omega \rightarrow [0,1]\). The crack phase-field function \(d(\textit{\textbf{x}})\) interpolates between \(d=1\), which indicates undamaged material, and \(d=0\), which indicates a fully broken material phase.

For stating the variational formulations, the spacesare used. Herein, \(W_{\mathrm{in}}\) denotes a closed, non-empty and convex set which is a subset of the linear function space \(W=\text {H}^1(\Omega )\) (see e.g., [29]).

In the following, a variational setting for quasi-brittle fracture in bulk materials with small deformations is formulated. To formulate the bulk free energy stored in the material, we define the first and second invariants aswith the second-order infinitesimal small strain tensor defined asThe isotropic scalar valued free-energy function readswhere \(K= \lambda +\frac{2}{3}\mu \) is the bulk modulus. A stress-free condition for the bulk energy-density function requires \({\widetilde{\Psi }}\left( I_1(\textit{\textbf{0}}),I_2(\textit{\textbf{0}})\right) =0\). Hence, the bulk free-energy functional including the stored internal energy and the imposed external energy iswhere \(\varvec{\tau }\) is the imposed traction traction vector on \(\partial _N\Omega _C := \Gamma _N \cup \mathcal {C}\) and the body-force is neglected.

Following [1], we define the total energetic functional which includes the stored bulk-energy functional and fracture dissipation aswhere \(G_c\) is the so called the Griffith’s critical elastic-energy release rate. Also, \(\mathcal {H}^{\delta -1}\) refers to the \((\delta -1)\)-dimensional Hausdorff measure (see e.g. [2]). Following [2], \(\mathcal {H}^{\delta -1}\) is regularized (i.e. approximated) by the crack phase-field \(d(\textit{\textbf{x}})\) (see e.g. [2]). Doing so, a second-order variational phase-field formulation is employed; see Sect. 2.3. Additional to that, a second-order stress degradation state function (intacted-fractured transition formulation) is used as a monotonically decreasing function which is lower semi-continuous order; see Sect. 2.5.

Let us represent a regularized (i.e., approximated) crack surface for the sharp-crack topology (which is a Kronecker delta function) thorough the exponential function \(d(\textit{\textbf{x}})=1-\exp ^{ - \vert \textit{\textbf{x}} \vert /l }\), which satisfies \(d(\textit{\textbf{x}})=0\) at \(\textit{\textbf{x}}=0\) as a Dirichlet boundary condition and \(d(\textit{\textbf{x}})=1\) as \(\textit{\textbf{x}}\rightarrow \pm \infty \). This is explicitly shown in Fig. 1b for different length scales. Here, \(\textit{\textbf{x}}\) is a position variable in the Cartesian coordinate system, meaning \(\textit{\textbf{u}}\) and d have a certain value at each position within the geometry. The first observation through the explicit formulation is that, the crack phase-field d constituting a smooth transition zone dependent on the regularization parameter \(\ell \). In engineering or physics, \(\ell \) is often a so-called characteristic length-scale parameter. This may be justified since this zone weakens the material and is a physical transition zone from the unbroken material to a fully damaged state. In practice, choices such as \(\ell =2h\) or \(\ell =4h\) are often employed. Following [30, 31], a regularized crack surface energy functional for the second term in Eq. 8 readsbased on the crack surface density function \(\gamma _{\ell }(d,\nabla d)\) per unit volume of the solid. This equation is the so-called AT-2 model because of the quadratic term in PDE.

We set sharp crack surfaces as Dirichlet boundary conditions in \(\mathcal {C}\subset \Omega \). Hence, the crack phase-field \(d(\textit{\textbf{x}},t)\) is obtained from the minimization of the regularized crack density function asFigure 2 gives the numerical solution that arises from the minimization Eq. 10 and demonstrates the effect of different regularized length scales on the numerical solution. Clearly, a smaller length scale leads to a narrower transition zone (see Fig. 2c). That is also in agreement with the crack phase-field profile shown in Fig. 1b.

Fracture mechanics is the process which results in the compression free state. As a result, a fracture process behaves differently in the positive phase and in negative phase, see e.g. [32]. In the following, an additive split for the strain energy density function to distinguish the positive and negative phases is used. Instead of dealing with a full linearized strain tensor \(\varvec{\varepsilon }(\textit{\textbf{u}})\), the additive decompositionof the strain tensor based on its eigenvalues is used [5, 31]. Herein, \(\langle x \rangle _{\pm } := \frac{ x {\pm } |x|}{2}\) refers to the a Macaulay brackets for \(x \in \mathbb {R}^\pm \). Furthermore, \(\varvec{\varepsilon }^{+}\) and \(\varvec{\varepsilon }^{-}\) refer to the positive and negative parts of the strain, respectively. The \(\{\varepsilon _i\}\) are the principal strains (i.e., the eigenvalues of the \(\varvec{\varepsilon }(\textit{\textbf{u}})\)) and the \(\{\mathbf{N }_i\}\) are the principal strain directions (i.e., the eigenvectors of the \(\varvec{\varepsilon }(\textit{\textbf{u}})\)). To determine the positive and negative parts of total strain \({\varvec{\varepsilon }}\), a positive-negative fourth-order projection tensor issuch that the fourth-order projection tensor \(\mathbb {P}^\pm _{\varvec{\varepsilon }}\) projects the total linearized strain \({\varvec{\varepsilon }}\) onto its positive-negative counterparts, i.e., \(\varvec{\varepsilon }^{\pm }=\mathbb {P}^\pm _{\varvec{\varepsilon }}:\varvec{\varepsilon }\). Hence an additive formulation of the strain-energy density function consisting of the positive and the negative parts readsHere, the scalar valued principal invariants in the positive and negative modes areHere, the first positive/negative invariant \(I_1(\varvec{\varepsilon })\) is strictly related to the tension/compression mode, respectively, meaning that if \(\mathrm{tr}(\varvec{\varepsilon })>0\) requires that we are in tension mode otherwise we are in compression state. The second invariant \(I_2(\varvec{\varepsilon })\) is mainly due to the positive and negative eigenvalue of the strain tensor, where its positive value requires that we are either in shear or in tension mode otherwise it is in compression. Thus, we distinguished between tension/compression and also a isochoric mode of our constitutive model, and only the positive part of the energy is degraded.

Due to the physical response of the fracture process, it is assumed that the degradation of the bulk material due to the crack propagation depends only on the tensile and isochoric counterpart of the stored bulk energy density function. Thus, there is no degradation of the bulk material in negative mode, see [31]. Hence, the degradation function denoted as \(g(d_+)\) acts only on the positive part of bulk energy given in Eq. 12, i.e.,This function results in degradation of the solid during the evolving crack phase-field parameter d. Due to the transition between the intact region and the fractured phase, the degradation function has flowing properties, i.e.,Following [31], the small residual scalar \(0<\kappa \ll 1\) is introduced to prevent numerical instabilities. It is imposed on the degradation function, which now readsThe stored bulk density function is denoted as \(w_{bulk}\). Together with the fracture density function \(w_{frac}\), it gives the the total density functionwith

Herein, to make sure that phase-field quantity d lies in the interval [0, 1], we define \(d_+\) to map negative values of d to positive values. In Formulation 2.1, the stationary points of the energy functional are determined by the first-order necessary conditions, namely the Euler–Lagrange equations, which can be found by differentiation with respect to \({\textit{\textbf{u}}}\) and d.

Herein, \({{\mathcal {E}}}_{\textit{\textbf{u}}}\) and \({{\mathcal {E}}}_d\) are the first directional derivatives of the energy functional \({{\mathcal {E}}}\) given in Formulation 2.1 with respect to the two fields, i.e., \(\textit{\textbf{u}}\) and d, respectively. Also, \({\widetilde{D}}\) is a crack driving state function which depends on a state array of strain- or stress like quantities and \(\delta \textit{\textbf{u}}\in \{ \mathbf{H}^1(\Omega )^2: \delta \textit{\textbf{u}}=\textit{\textbf{0}} \; \mathrm {on} \; \partial \Omega _D \}\) is the deformation test function and \(\delta d\in H^1(\Omega )\) is the phase-field test function.

Furthermore, the second-order constitutive stress tensor with respect to Eq. 18 readswith

Following [33, 34], we determine the crack driving state function to couple between two PDEs. Hence, crack driving state function acts as a right hand side for the phase-field equation. To formulate the crack driving state function, we consider the crack irreversibility condition, which is the inequality constraint \(\dot{d} \le 0\) imposed on our variational formulation. The first variation of the total pseudo-energy density with respect to the crack phase-field given in (17) readsHerein, the functional derivative of \(\gamma _l(d,\nabla d)\) with respect to d isMaximization the inequality given in Eq. 22 with respect to the time history \(s\in [0,t_n]\) readsWe multiply Eq. 24 by \(\frac{l}{G_c}\). Then Eq. 24 can be restated asHere, \(\mathcal {H}:=\mathcal {H}(\varvec{\varepsilon },t)\) denotes a positive crack driving force that is used as a history field from initial time up to the current time. Note that the crack driving state function \({\widetilde{D}}\) is affected by the length-scale parameter \(\ell \) and hence depends on the regularization parameter.

The multi-field problem given in Formulation (2.3) depending on \(\textit{\textbf{u}}\) and d implies alternately fixing \(\textit{\textbf{u}}\) and d, which is a so called alternate minimization scheme, and then solving the corresponding equations until convergence. The alternate minimization scheme applied to the Formulation (2.3) is summarized in Algorithm 1.

In this part, the influence of the \(\kappa \) on the stress-strain curve is taken into account. Following [26], the homogeneous solution at the quasi-static stationery state of the phase-field partial differential equation in the loading case takes the following formwhich results from the free Laplacian operator \(\Delta (\bullet )=0\) assumption in Eq. 24 without any source terms (zero left-hand sides). Here, the crack driving state function \({\widetilde{D}}\) is given in Eq. 25. Because, we are in the elastic limit, prior to the onset of fracture, then no split is considered. We now aim to relate a stress state \(\sigma \) with the isotopic phase-field formulation. To do so, a non-monotonous function in the one-dimensional setting for the degrading stresses takes the following form byTo see the influence of the \(\kappa \) in Eq. 29, the concrete material is considered. Following, [35] for a concrete material which has a brittle response, a typical values for material parameters reads,We set \(\ell =0.0105~\mathrm{m}\). Thus, we can do a plot for the stress-strain curve through Eq. 29 by considering the material set given above. Figure 3 shows the effect of stress state for different strain loading. The black curve represents the stress-strain curve while \(\kappa =0\). Evidently, it can be grasped through Fig. 3 with \(\kappa =0\) the \(\sigma _c\) is exactly \(\sigma _c=4.5\,\mathrm{MPa}\) as it is required for the concrete material, see [35]. If we consider \(\kappa \ne 0\) as a function of characteristic length-scale, see Fig. 3 left, we can observe a good agreement with \(\kappa =0\) up to the peak point while after some strain value it becomes different as \(\kappa \) changes. Unfortunately, we can not observe any converged response if we consider \(\kappa \) as a function of \(\ell \). In contrast, if we chose \(\kappa \) sufficiently small, see Fig. 3 right, as much as \(\kappa \) reduced, in here less than \(\kappa \le 10^{-4}\), we observed a very identical response with \(\kappa =0\), thus it behaves as numerical parameters rather than material parameters.

We consider Formulation 2.3 as the forward model \(\mathbf{y} =\mathcal {G}(\,\Theta \,(\textit{\textbf{x}}))\), where \(\mathcal {G}:L^2(\Omega )\rightarrow L^2(\Omega )\). The forward model explains the response of the model to different influential parameters \(\Theta \) (here \(\mu \), K, and \(G_c\)). We can write the statistical model in the form [37]where \(\mathcal {M}\) indicates a vector of observations (e.g., measurements). The error term \(\varepsilon \) arises from uncertainties such as measurement error due experimental situations. More precisely, it is due to the modeling and the measurements and is assumed to have a Gaussian distribution of the form \(\mathcal {N}(0,H)\) with known covariance matrix H. The error is independent and identically distributed and is independent from the realizations. Here, for sake of simplicity, we assume \(H=\sigma ^2I\) (for a positive constant \(\sigma ^2\)).

For a realization \(\theta \) of the random field \(\Theta \) corresponding to a realization m of the observations \(\mathcal {M}\), the posterior distribution is given bywhere \(\pi _0(\theta )\) is the prior density (prior knowledge) and \({W_m}\) is the space of parameters m (the denominator is a normalization constant) [38]. The likelihood function can be defined as [37]As an essential characteristic of the phase-field model, the load-displacement curve (i.e., the global measurement) in addition to the crack pattern (i.e., the local measurement) are appropriate quantities to show the crack propagation as a function of time. Figure 4 indicates the load-displacement curve during the failure process. Three major points are the following.

① First stable position. This point corresponds to the stationary limit such that we are completely in elastic region (\(d(\textit{\textbf{x}},0)=1 \;\forall \textit{\textbf{x}}\in \Omega \backslash \mathcal {C}\)).

② First peak point. Prior to this point crack nucleation has occurred and now we have crack initiation. Hence, this peak point corresponds to the critical load quantity such that the new crack surface appears (i.e., there exist some elements which have some support with \(d=0\)).

The interval between point 1 and point 2 in Fig. 4 typically refers to the primary path where we are almost in the elastic region. The secondary path (sometimes referred to as the softening damage path) starts with crack initiation occurring at point 2. The whole process recapitulates the load-deflection curve in the failure process.

The main aim of solving the inverse problem followed here is to determine the random field \(\Theta \) to satisfy (38). We strive to find a posterior distribution of suitable values of the parameters \(\mu \), K, and \(G_c\) in order to match the simulated values (arising from (27)) with the observations. The distribution provides all useful statistical information about the parameter.

The crack pattern is a time-dependent process (more precisely in a quasi-static regime, the cracking process is load-dependent), i.e., after initiation it is propagated through time. In order to approximate the parameters precisely, we estimate the likelihood during all time steps. Therefore, the posterior distribution maximizes the likelihood function for all time steps, and therefore we have an exacter curve for all crack nucleation and propagation times.

MCMC is a suitable technique to calculate the posterior distribution. When the parameters are not strongly correlated, the MH algorithm [39] is an efficient computational technique among MCMC methods. We propose a new candidate (so-called \(\theta \), i.e., a value of \((\mu ,{K},G_c)\)) according to a proposal distribution (for instance uniform or normal distributions) and calculate its acceptance/rejection probability. The ratio indicates how likely the new proposal is with regard to the current sample. In other words, by using the likelihood function (40), the ratio determines whether the proposed value is accepted or rejected with respect to the observation (here the solution of Formulation 2.3 with a very fine mesh). As mentioned, fast convergence means that the parameters are fully correlated. A summary of the MH algorithm is given below.

This example considers the single edge notch tension. The specimen is fixed at the bottom. We have traction-free conditions on both sides. A non-homogeneous Dirichlet condition is applied at the top. The domain includes a predefined single notch (as an initial crack state imposed on the domain) from the left edge to the body center, as shown in Fig. 5a. We set \(A=0.5\;\mathrm {mm}\) hence \(\Omega =(0,1)^2 \mathrm {mm}^2\), hence the predefined notch is in the \(y=A\) plane and is restricted to \(0\le |\mathcal {C}|\le A\). This numerical example is computed by imposing a monotonic displacement \({\bar{u}}=1\times 10^{-4}\) at the top surface of the specimen in a vertical direction. The finite element discretization corresponding to \(h=1/80\) is indicated in Fig. 5b.

For the shear modulus, we assume the variation range \((60\,\mathrm {kN/mm^2},100\,\mathrm {kN/mm^2})\). Regarding the the bulk modulus K, the parameter varies between \(140\,\mathrm {kN/mm^2}\) and \(200\,\mathrm {kN/mm^2}\). Finally, we consider the interval between \(2.1\times 10^{-3}\,\mathrm {kN/mm^2}\) and \(3.3\times 10^{-3}\,\mathrm {kN/mm^2}\) for \(G_c\). Furthermore, we assume that in this example, the variables are spatially constant random variables (they are not random fields).

We solved the PDE model (Formulation 2.3) with \(\mu =80\,\mathrm {kN/mm^2}\), \(K=170\,\mathrm {kN/mm^2}\), and \(G_c=2.7\times 10^{-3}\,\mathrm {kN/mm^2}\) [31] and the displacement during the time (as the reference solution) with \(h=1/320\) was obtained. The main goal is to obtain the suitable values of \(\mu ,~{K}\), and \(G_c\) such that the simulations match the reference value.

For this example, we use a uniformly distributed prior distribution and the uniform proposal distributionwhere \(\chi \) indicates the characteristic function of the interval \([\theta _1,\theta _2]\) (where \(\theta \) denotes a set of parameters).

First, we describe the effect of each parameter on the displacement. As the elasticity constants (i.e., \(\mu \) and \({K}\)) become larger, the material response becomes stiffer; crack initiation takes longer to occur. Additionally, a larger crack release energy rate (as an indicator for the material resistance against the crack driving force) delays crack nucleation and hence crack dislocation. All these facts are illustrated in Fig. 6.

The joint probability density of the elasticity parameters and the marginal probability of the posterior are shown in Figure 7 including one- and three-dimensional Bayesian inversions. The mean values of the distributions are \(\mu =84.2\,\mathrm {kN/mm^2}\) and \(\mu =85.1\,\mathrm {kN/mm^2}\) are obtained for the shear modulus. Here an acceptance rate of 27% is obtained. Regarding the material stiffness parameter \(G_c\), the acceptance rates are near 29%. The values are summarized in Table 1.

To verify the parameters obtained by the Bayesian approach, we solved the forward model using the mean values of the posterior distributions. Figure 8 shows the load-displacement diagram according to prior and posterior distributions. As expected, during the nucleation and propagation process, using Bayesian inversion results in better agreement (compared to the prior). Furthermore, a better estimation is achieved by simultaneous multi-dimensional Bayesian inversion. From now onward, this approach will be used for Bayesian inference.

This numerical example is a fracture process that occurs through the coalescence and merging of two cracks in the domain. We consider the tension test with a double notch located on the left and right edge. The specimen is fixed on the bottom. We have traction-free conditions on both sides. A non-homogeneous Dirichlet condition is applied to the top-edge. The domain has a predefined two-notch located in the left and right edge in the body as shown in Fig. 15a. We set \(A:=20\,\mathrm {mm}\) and \(B:=10\,\mathrm {mm}\) hence \(\Omega =(20,10)^2\,\mathrm {mm^2}\). For the double-edge-notches, let \(H_1:=5.5\,\mathrm {mm}\) and \(H_2:=3.5\,\mathrm {mm}\) with the predefined crack length of \(l_0:=5\,\mathrm {mm}\) (Fig. 15a). This numerical example is computed by imposing a monotonic displacement \({\bar{u}}=1\times 10^{-4}\) at the top surface of the specimen in a vertical direction. The finite element discretization that uses \(h=1/80\) is indicated in Fig. 15b.

According to the truncated KL-expansion, for the bulk modulus K, Eq. 37 gives the mean value of \({\bar{K}}=23.58\) and the standard deviation of \(\sigma _{K}=0.28\). Therefore, the parameter varies between \(10\,\mathrm {kN/mm^2}\) and \(14\,\mathrm {kN/mm^2}\). For the shear modulus, the expectation of \( \bar{\mu }=22.8\) and the standard deviation of \(\sigma _\mu =0.23\) leads to the variation range \((6\,\mathrm {kN/mm^2},10\,\mathrm {kN/mm^2})\). Similarly, by using a KL-expansion for \(G_c\), we obtained the variation range between \(8\times 10^{-5}\,\mathrm {kN/mm^2}\) and \(12\times 10^{-5}\,\mathrm {kN/mm^2}\). Figure 16 illustrates the effect of their different values including shear and bulk modulus and \(G_c\) on the curve.

We assumed the uniform proposal distribution, namely the normal distributionAs we aforementioned, the random field can be represented using the KL-expansion. In this example (32) is employed to parameterize the elasticity and energy rate parameters. The random perturbations are imposed on the \(\xi \) coefficients in the KL expansion. According to the proposal, the mean of the KL-expansion is updated.

Here we plan to study the effect of the number of samples N on the posterior distribution. Figure 17 shows the joint distributions of (\({K}\), \(\mu \)), and the marginal distribution of \(G_c\) using \(N=3\,000\), \(N=12\,000\), and \(N=50\,000\). The calculations are done with \(h=1/80\), and \(h=1/320\) is used as the reference. As shown, with a larger number of samples, the distribution is close to a normal distribution. Table 4 points out the mean values in addition to the acceptance rate of all influential parameters.

As the next step, we use different mesh sizes for the Bayesian inversion using 15 000 samples. Figure 18 shows the crack pattern using different mesh sizes changing from \(h=1/10\) to \(h=1/320\). Finer meshes lead to a smoother and more reliable pattern. Figure 19 depicts the load-displacement diagram using the prior values. With coarse meshes, the curve is significantly different from the reference including crack initiation. Using Bayesian inversion (see Fig. 17) enables us to predict the crack propagation and initiation more precisely. As the figure shows, even for the coarsest mesh (compare \(h=1/10\) to \(h=1/320\)) the peak and fracture points are estimated precisely. For finer meshes (e.g., \(h=1/80\)) the diagram is adjusted tangibly compared to the reference value. Finally, a summary of the mean values (of posterior distributions) and their respective acceptance rate is given in Table 5.

The significant advantage of the developed Bayesian inversion is a significant computational cost reduction. As shown, for SENT and DENT, by using Bayesian inference for coarser meshes, the estimated load-displacement curve is very close to the reference values. We should note that the needed CPU time for \(h=1/320\) is approximately 4.5 hours; however the solution with \(h=1/80\) is obtained in less than 10 minutes. This fact pronounces the computational efficiency provided by Bayesian inversion, i.e., obtaining a relatively precise solution in spite of using much coarser meshes.

Here we consider the tension test where two voids are located in the domain as a more complicated example. The voids are used to weaken the material and to lead to crack nucleation/initiation without an initial singularity (i.e., a pre-existing crack). The specimen is fixed on the bottom. We have traction-free conditions on both sides. A non-homogeneous Dirichlet condition is applied to the top. Domain includes a predefined two voids in the body, as depicted in Fig. 20a. We set \(A=0.5\,\mathrm {mm}\) hence \(\Omega =(0,1)^2\,\mathrm {mm^2}\). The radius of left void is \(r_1:=0.247 \) with the center \(c_1:=(0.21,0.197)\). The radius of the right void is \(r_2:=0.0806\) with the center \(c_2:=(0.7,0.197)\). This numerical example is computed by imposing a monotonic displacement \({\bar{u}}:=1\times 10^{-4}\) at the top surface of the specimen in vertical direction. The finite-element discretization corresponding \(h=1/40\) is shown in Fig. 20b.

Due to the resemblance to the first example (SENT) we use the same range of parameters and the variables are again spatially constant random variables. The load-displacement curves obtained from different values of \(\mu \), \({K}\), and \(G_c\) are illustrated in Fig. 21.

This numerical example includes two voids results in multi-stage crack propagation. Hence, in the load-displacement curve, two peak points exist to demonstrate multi-stage crack propagation, see Fig. 21.

Figure 22 shows the proposal distribution where a uniform prior distribution is used for Bayesian inversion with 10 000 samples. Here we use \(h=1/160\) as the reference solution and \(h=1/80\) is employed to estimate the parameters. In summary, the mean values are \(\mu =63\,\mathrm {KN/mm^2}\), \(K=162\,\mathrm {KN/mm^2}\), and \(G_c=0.003\,\mathrm {KN/mm^2}\), and the acceptance rates are 28% (the elasticity parameters) and 21% (the critical energy rate).

We solve the forward model with the mean values of the estimated parameters. As Fig. 23 shows, the difference between the prior distribution and the reference solution is significantly large. By using Bayesian inversion, we could compensate this difference; crack initiation and material failure points are estimated precisely. Although multidimensional Bayesian inversion increases the computational costs (CPU time), the estimated solution is closer to the reference value.

Finally, we show the crack patterns obtained by different meshes varying from \(h=1/10\) to \(h=1/160\). We use Bayesian inference to estimate the unknown parameters with the three-dimensional approach for \(h=1/20\) and \(h=1/40\). As Fig. 25 illustrates, although the solution based on the posterior distribution is more precise (i.e., a better estimation of crack initiation and the fracture point) compared to the one based on the prior distribution, there is still a difference compared to the reference value. These results conform to Fig. 24, since the estimated crack pattern is considerably larger than the reality.