esa-sceva/satcom-chunk-collection huggingface dataset:
# Bearing Capacity of Ring Foundations on Sand Overlying Clay
## 4 Comparisons of Previous Studies  
### For Circular Footings on Homogeneous Clay  
In Table 1, the bearing capacity factor (\(N_{c}\)) for a rough circular footing on homogeneous clay obtained from the present analysis is compared with existing solutions. Regarding the results of the theoretical exact solution reported by [PERSON] [44], and the method of characteristics performed by [PERSON] and [PERSON] [45], [PERSON] and [PERSON] [46], and [PERSON] [47], it can be seen that the values for \(N_{c}\) lie between lower and upper bounds predicted by the present analysis, except for the values obtained by [PERSON] and [PERSON] [45]. The present values for \(N_{c}\) correspond very closely to the semi-empirical solutions reported by [PERSON] [48], [PERSON] [3], and [PERSON] [4] and to the finite element methods conducted by [PERSON] et al. [49], [PERSON] et al. [50], [PERSON] and [PERSON] [51], and [PERSON] et al. [33]. The upper bound solutions of [PERSON] et al. [52] and [PERSON] et al. [43] provide slightly greater values of \(N_{c}\), compared to the upper bound of the present analysis. The lower bound values for \(N_{c}\), obtained by [PERSON] et al. [43], [PERSON] and [PERSON] [53], and [PERSON] and [PERSON] [30] were found to be marginally smaller than the lower bound of the present study.  
\begin{table}
\begin{tabular}{c c c c} \[PERSON] (1951) \({}^{\mathrm{a}}\) & 6.05 & [PERSON] et al. (2004) \({}^{\mathrm{c}}\) & 6.227 \\ [PERSON] (1951) \({}^{\mathrm{b}}\) & 6.00 & [PERSON] et al. (2005) \({}^{\mathrm{f}}\) & 6.09 \\ [PERSON] (1970) \({}^{\mathrm{b}}\) & 6.17 & [PERSON] et al. (2006) \({}^{\mathrm{f}}\) & 5.96 \\ [PERSON] (1973) \({}^{\mathrm{b}}\) & 6.17 & [PERSON] and [PERSON] (2009) \({}^{\mathrm{e}}\) & 6.00 \\ [PERSON] et al. (1986) \({}^{\mathrm{c}}\) & 6.31 & [PERSON] and [PERSON] (2010) \({}^{\mathrm{f}}\) & 6.166 \\ [PERSON] and [PERSON] (1995) \({}^{\mathrm{d}}\) & 6.34 & [PERSON] and [PERSON] (2015) \({}^{\mathrm{e}}\) & 6.00 \\ [PERSON] and [PERSON] (2003) \({}^{\mathrm{d}}\) & 6.05 & [PERSON] et al. (2016) \({}^{\mathrm{f}}\) & 6.08 \\ [PERSON] (2004) \({}^{\mathrm{d}}\) & 6.05 & Present study \({}^{\mathrm{e}}\) & 6.025 \\ [PERSON] et al. (2004) \({}^{\mathrm{e}}\) & 5.856 & Present study \({}^{\mathrm{c}}\) & 6.058 \\ \hline \end{tabular} Note: \({}^{\mathrm{a}}\) exact solution; \({}^{\mathrm{b}}\) semi-empirical solution; \({}^{\mathrm{c}}\) upper bound solution; \({}^{\mathrm{d}}\) method of characteristics; \({}^{\mathrm{e}}\) lower bound solution; \({}^{\mathrm{f}}\) finite element method.  
\end{table}
Table 1: Comparison of bearing capacity factor (\(N_{c}\)) for a rough circular foundation on homogeneous clay.
## 3 Results and Discussion
### Nanomechanical Properties of Electrospun PLLA Nanofibers
#### 3.4.4 Comparison of the Elastic Modulus between Pristine and Annealed Nanofibers  
The overlapping of the loading and unloading curves in PFQNM and the absence of hysteresis in the F\({}_{\text{I}}\) curves in ImAFM are two pieces of evidence proving that there is neither plastic deformation nor viscoelastic behavior in both as-spun and annealed PLLA nanofibers. Furthermore, the calculation of Maugis parameters leads to the conclusion that the DMT model is the best fit model for both materials. With all these considerations, we can thus apply the DMT model to extract the modulus value. Figure 6 shows the elastic modulus as a function of the fiber diameter for each case. It appears that the smaller the diameter of a nanofiber is, the higher the elastic modulus it has. This size effect on the elastic modulus of nanofibers has been reported in numerous studies. It has been proved that orientation in crystalline and amorphous domains are higher in thinner fibers, resulting in an increase of elastic modulus [51; 52; 53]. Since tapping mode AFM measurements showed that there is a reduction of about 30% in diameter of nanofibers from the annealing process (from \(210\pm 88\) nm for the pristine fibers to  
Figure 5: ImAFM phase (°), amplitude (nm) of as-spun (**a**,**b**) and annealed (**c**,**d**) PLLA nanofibers, respectively; (**e**) \(F_{I}\) and \(F_{Q}\) curves of those samples taken at the black crosses. The black \(F_{I}\) and \(F_{Q}\) curves correspond to one pixel from a PS domain of the reference sample (see Supporting Information).  
\(160\pm 44\) nm for the annealed samples), for a meaningful comparison, we should compare the elastic modulus of annealed nanofibers to that of as-spun nanofibers with a relatively higher diameter. For example, we should compare the elastic modulus of a 200-nm pristine fiber with a 170-nm annealed fiber, and so on. In this regard, the nanomechanical properties of the annealed fibers are definitely higher than those of untreated ones. The increase in elastic modulus of annealed nanofibers can be explained by the increase in the crystallinity upon annealing. The surface of annealed nanofibers consists of many highly packed, granular domains which, in turn, enhance the surface stiffness of the sample under applied stress normal to the fiber axis. It should also be noted that the elastic modulus measured by PFQNM is in the direction perpendicular to the fiber axis so these values should be expected to be different from values measured by classical methods, such as the three-point bending test, which returns the axial modulus values of the fiber.  
However, in our study, we are more interested in how the nanofibers respond to stress normal to the fiber axis because it also corresponds to the piezoelectric axis. In this case, corresponding to higher elastic modulus in the normal axis, annealed nanofibers can withstand higher stress and higher deformation, which links directly to higher energy output for a piezoelectric energy-harvesting device.
# A Robust and Secure Watermarking Approach Based on Hermite Transform and SVD-DCT
## 5 Experiments and Results
### Performance Measures  
To assess the performance of our watermarking algorithm, we employed the following metrics [53]: Mean-Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), Mean Structural Similarity Index (MSSIM), Normalized Cross-Correlation (NCC), Bit Error (\(B_{error}\)), and Bit Error Rate (BER). In the cases of the MSE, PSNR, SSIM, MSSIM, and NCC, the images were \(X\times Y\) px, where \(x\) and \(y\) represent the spatial coordinates:  
* The MSE refers to a statistical metric used to measure the image's quality. It evaluates the squared difference between a pixel in the original image \(I_{1}\) and the watermarked image \(I_{2}\). After calculating this result for all pixels in the image, it returns the average result, as is shown in Equation (23): \[MSE=\frac{1}{XY}\sum_{x=1}^{X}\sum_{y=1}^{Y}\Big{[}I_{1}(x,y)-I_{2}(x,y)\Big{]} ^{2}.\] (23) If the MSE = 0, this indicates that there is no error between the original image and the watermarked image.
* The PSNR is defined in Equation (24): \[PSNR=10\log_{10}\bigg{(}\frac{255^{2}}{MSE}\bigg{)}.\] (24) On the one hand, a higher image quality is achieved with a higher PSNR value, so it approaches infinity. On the other hand, low PSNR values report high differences between the images [57].
* The Mean Structure Similarity Index (MSSIM) has a value determined by Equation (25): \[MSSIM(I,\hat{I})=\frac{1}{M}\sum_{j=1}^{M}SSIM(I_{j},\hat{I}_{j}),\] (25) where \(I\) and \(\hat{I}\) correspond to the original and the distorted image, respectively, \(I_{j}\) and \(\hat{I}_{j}\) represent their \(j\)-th local window, and \(M\) stands for the amount of local windows of the image. In the case that \(I_{j}\) and \(\hat{I}_{j}\) have no negative values, the value of the SSIM is calculated as shown in Equation (26): \[SSIM(I,\hat{I})=\frac{(2\mu_{I}\mu_{I}+C_{1})(2\sigma_{I\hat{I}}+C_{2})}{(\mu_ {I}^{2}+\mu_{I}^{2}+C_{1})(\sigma_{I}^{2}+\sigma_{I}^{2}+C_{2})},\] (26) where the averages of \(I\) and \(\hat{I}\) are given by \(\mu_{I}\) and \(\mu_{I}\), respectively, their standard deviations are given by \(\sigma_{I}\) and \(\sigma_{I}\), the covariance between both images is represented by \(\sigma_{I\hat{I}}\), and the constants \(C_{1}\) and \(C_{2}\) are used to prevent instability if the denominator happens to have a value close to zero [58].
## 4 Results
### Time for Tie Points Detection and Matching  
The first test was an analysis of the average time for detection and matching of characteristic points on all pairs of raster images in the various projections for both test sites (Table 2) in minutes and seconds.  
It can be seen from Table 2 that the shortest time for searching and matching of characteristic points for all algorithms is obtained in the case of both test sites for representation of the point cloud in the form of an orthoimage; longer times are obtained for the Mercator projection and the spherical image, respectively. Regarding computation times for all analysed cases, the shortest time is obtained when the FAST detector is used.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Average Pre-Processing Time [Minutes \& Seconds]**} \\ \hline \multicolumn{6}{c}{**Test Site I**} & \multicolumn{3}{c}{**Test Site II**} \\ \hline \multicolumn{6}{c}{**Cartographic Transformation**} \\ \hline Spherical & Orthoimage & Mercator & Spherical & Orthoimage & Mercator \\
11 \& 15 & 13\&10 & 14\&51 & 11\&56 & 15\&25 \\ \hline \multicolumn{6}{c}{**Test Site III**} & \multicolumn{3}{c}{**Test Site IV**} \\ \hline
11\&5 & 17\&20 & 15\&01 & 11\&36 & 15\&31 &15\&09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The pre-processing data time.  
Figure 9: The example of the point cloud in the spherical projection of test site IV: “Empty shop (shopping mall)” with marked check points (red circles).
## 3 Method
### Certificate Graph Convolutional Networks (Cert GCN)  
Since we wish to use the graph structure to represent the certificate dataset, which contains many discrete pem structure files, all of these certificate documents are packaged into a corpus of certificates, represented in graphical structure. To facilitate the study, this paper gives a formal definition of the structure of a certificate graph, which consists of edges of certificate nodes and certificate nodes.  
**Theorem 1**.: \(G_{\text{cert}}\) _is a certificate graph that is composed of nodes, edges, and the adjacency matrix. It is formulated as \(G_{\text{cert}}=(V,E,\widetilde{A})\), where \(V=(V_{\text{attribute}},V_{\text{document}})\) and \(\widetilde{A}=A+I\)._  
_Where \(V_{\text{attribute}}\) is the set of certificate attributes, and \(V_{\text{document}}\) is the set of document names. \(E\) represents the relationship between nodes (document-to-attribute and attribute-to-attribute). The certificate graph structure is described in terms of the adjacency matrix \(\widetilde{A}\) where \(A\) is constructed from the relationships between nodes and edges, and \(I\) stands for the identity matrix, which is added for each node to solve the problem of its own information loss._  
We detect malicious certificates using a GCN. To facilitate the study of GCNs for the detection of malicious digital certificates, we define the formula for Cert GCN.  
**Theorem 2**.: _Cert GCN refers to a GCN based on certificate documents and is used for malicious digital certificate detection. \(Z\) represents a two-layer Cert GCN, where \(Z=linear(BReLU(BXW_{0})W_{1})\)._  
_Where \(B=D^{-\frac{1}{2}}\widetilde{A}D^{-\frac{1}{2}}\), \(\widetilde{A}\) is the adjacency matrix mentioned in Theorem 1, \(D\) is the degree matrix of \(A\) (\(D_{ii}=\sum_{j}A_{ij}\)), and \(X\) is the input layer representation feature matrix. Each node \(i\) has feature \(x_{i}\). The characteristic matrix of a node can be represented by the matrix \(X_{N\times M}\), where \(N\) is the number of nodes and \(M\) denotes the number of features per node. \(W_{0}\) is the weight matrix for the first layer and \(W_{1}\) is the weight matrix for the second layer. The first layer of the activation function is ReLU. The second layer is the linear activation function._  
In this paper, we constructed a dataset of certificate graph structures and put the dataset into the GCN layer. Each layer of the network consists of a convolutional layer, an activation function layer, and a dropout layer. The first layer of the activation function is _ReLU_. The second layer is the linear activation function. The constructed Cert GCN model is shown in Figure 2.  
The flowchart for malicious digital certificate detection is shown in Figure 3.  
Figure 2: Schematic of Cert GCN.
## 2 Results and Discussion
### Gene Ontology Enrichment Analyses of PbPP1c-Interacting Proteins  
In order to gain some insight into potential pathways in which PP1c could be involved and to avoid bias in their identification, GO enrichment pathways including all detected proteins in each parasite stage were explored using PlasmoDB software (release 51, March 2021). We analyzed enriched GO terms/pathways of computed and curated biological processes. In the case of schizons, 37 pathways were found to be enriched, including 5 with a strong fold enrichment (>5) involved in spliceosomal SnRNP assembly, formation of cytoplasmic translation initiation complex, cytoplasmic translation, ribonucleoprotein complex assembly, and subunit organization (Supplementary Materials Table S2). Closer examination of GO-enriched terms revealed that a large list of genes had multiple annotations and, in some cases, GO pathways included shared or the same set of genes, leading to overlap and duplication. Consequently, and in the search for the most informative data, full GO terms were reduced using the GO slimmer tool (PlasmoDB release 51) to obtain Slim annotations that were further inspected to reassign, when appropriate, to GO parent terms (Table 1). However, general GO terms were excluded (such as biosynthetic process and metabolic process) while branch information was retained to ensure that the enriched family of each function was present in the data. The results showed that, in schizont stages, PbPP1c was predominantly associated with \"cellular component assembly\" (20 proteins), encompassing \"protein-containing complex assembly\" (18 proteins), \"ribonucleoprotein assembly\" (13 proteins), and in \"translation\" complex (57 proteins) (Table 1). Next, the proteins involved in each main GO Slim pathways underwent the retrieval of networks for interacting proteins using STRING software (v:11.0b) combined with Cytoscape visualization (v.3.8.2, accessed in July 2021). In the \"cellular component assembly\" pathway, the results presented in Figure 3 showed a protein-protein interaction network formed of 19 nodes and 53 edges. Three KEGG pathways were significantly enriched in this network (Figure 3). This includes proteasome and spliceosome complexes, confirming the previous analysis [14] and suggesting that these proteins may at least constitute PP1 substrates. In fact, it has been shown that PP1c was able to inhibit human proteasome activities in vitro [24]. In the case of spliceosomes, PP1c has been shown to be an essential player in splicing by contributing to the dephosphorylation of some spliceosome factors [25]. Interestingly, the analysis of data from our study on the PbPP1c interactome shows some overlap with a recent study examining global protein complexes in schizons from _P. falciparum_, _P. knowlesi_, and _P. berghei_ through native-PAGE fractionation coupled with spectrometry and machine learning [26]. This study allowed the identification of >20,000 putative protein-protein interactions, organized in 600 clusters. The reanalysis of these data revealed the presence of 92 putative PP1c interactors
# A Cryptography-Powered Infrastructure to Ensure the Integrity of Robot Workflows
## 3 Compliance Architecture
### Adding Workflow Verification to an Execution Entity
#### 3.3.2 Execution Controller  
The execution controller enables the execution of workflow queues on systems that do not support this out of the box. As an example, on a robot system where only a single program can be triggered, the execution controller will orchestrate the subsequent execution of multiple programs. This feature is required to enable the bulk signature of larger programs. Otherwise, it would be required to sign each workflow individually potentially requiring each robot movement to carry a specific signature. This would not only be tedious and cause additional overhead but also the overall integrity of the program could no longer be ensured meaning that single steps or workflows could be re-ordered without being noticed.
## Appendix A  
\[\left\{\begin{array}{ll}\frac{d\phi(t)}{dt}=b(t)(\phi(t)-\psi(t))&\text{( A1)}\\ \frac{d\phi(t)}{dt}=\mu(t)\phi(t)&\text{(A2)}\\ b(t)=\frac{b}{1+\beta\exp(-bt)}&\text{(A3)}\end{array}\right.\]  
Equation (A2) can be expanded as follows [45; 46]:  
\[\frac{d\phi(t)}{dt}=\{\mu(t)+\sigma\gamma(t)\}\phi(t)\] (A4)  
We expand Equation (A4) with \(It\,\widetilde{\,\,}\)'s formula [45; 46]:  
\[d\phi(t)=\left\{\mu(t)+\frac{1}{2}\sigma^{2}\right\}\phi(t)dt+\sigma\phi(t)d \eta(t)\] (A5)  
where \(\eta(t)\) is a one-dimensional Wiener process which is a Gaussian distribution. The properties of the Wiener process are as follows:  
\[\Pr[\eta(0)=0]=1\] \[E[\eta(t)]=0\] \[E[\eta(t)\eta(t^{\prime})]=Min[t,t^{\prime}]\]When \(t=0\), \(\phi(t)=a\). We can derive the following solution of Equation (A5) by using \(\widehat{I\phi}\) type formula.  
\[\phi(t)=a\exp(\int_{0}^{t}\mu(x)dx+\sigma\eta(t))\] (A6)  
We assumed that the intensity function of the fault introduction obeys the Weibull distribution.  
\[\int_{0}^{t}\mu(x)dx=(1-\exp(-\theta t^{d}))\] (A7)  
Substituting Equation (A7) into Equation (A6), the density function of \(\eta(t)\) is defined as follows:  
\[f(\eta(t))=\frac{1}{\sqrt{2\pi t}}\exp\{-\eta(t)^{2}/2t\}\] (A8)  
We can derive the following expected value from Equation (A6):  
\[E[\phi(t)]=a\exp(1-\exp(-\theta t^{d})+\frac{1}{2}\sigma^{2}t)\] (A9)  
Given \(D(t)=\int_{0}^{t}b(t)dt\), then \(\exp(D(t))=\frac{\beta+\exp(bt)}{1+\beta}\). Equation (A1) is multiplied by \(\exp(D(t))\) on both sides:  
\[\exp(D(t))d\psi(t)+b(t)\exp(D(t))\psi(t)dt=b(t)\exp(D(t))\phi(t)dt\] (A10)  
The two sides of Equation (A10) are integrated:  
\[\int\exp(D(t))d\psi(t)+b(t)\exp(D(t))\psi(t)dt=\int b(t)\exp(D(t))\phi(t)dt\] (A11)  
\[\begin{array}{l}\exp(D(t))\psi(t)=\int b(t)\exp(D(t))\phi(t)dt\\ \psi(t)=\exp(-D(t))\int\phi(t)d(\exp(D(t)))\\ =\exp(-D(t))(\phi(t)\exp(D(t))-\int\exp(D(t))d(\phi(t))\\ =\phi(t)-\exp(-D(t))\int\exp(D(t))d(\phi(t))\\ =\frac{\phi(t)}{1+\beta\exp(-bt)}-\frac{\int\exp(bt)d(\phi(t))}{\beta+\exp(bt)} \end{array}\] (A12)  
To simplify the calculation, we expand the following formula with Taylor formula:  
\[\exp(-\theta t^{d})\approx 1\] (A13)  
\[t^{d-1}\approx 0\] (A14)  
Substituting Equations (A13) and (A14) into Equation (A9),  
\[d\phi(t)=a\exp(\frac{1}{2}\sigma^{2}t)\frac{1}{2}\sigma^{2}dt\] (A15)  
Substituting Equation (A15) into Equation (A12),  
\[\begin{array}{l}\psi(t)=\frac{\phi(t)}{1+\beta\exp(-bt)}-\frac{\sigma^{2}( \exp((b+\frac{1}{2}\sigma^{2})t)+\mathbb{C})}{(\beta+\exp(bt))(2b+\sigma^{2}) }\\ =\frac{a\exp(1-\exp(-bt^{d})+\frac{1}{2}\sigma^{2}t)}{1+\beta\exp(-bt)}-\frac{ \omega^{2}(\exp((b+\frac{1}{2}\sigma^{2})t)+\mathbb{C})}{(\beta+\exp(bt))(2b+ \sigma^{2})}\end{array}\] (A16)  
when \(t=0\), \(\psi(t)=0\) and \(\phi(t)=a\), then  
\[C=\frac{2b+\sigma^{2}}{\sigma^{2}}-1\] (A17)  
Substituting Equation (A17) into Equation (A16),\[\psi(t)=\frac{a\exp(1-\exp(-\theta t^{d})+\frac{1}{2}\sigma^{2}t)}{1+\beta\exp(- bt)}-\frac{a\sigma^{2}\exp(-bt)(\exp((b+\frac{1}{2}\sigma^{2})t)+\frac{2b+\sigma^{2} }{\sigma^{2}}-1)}{(1+\beta\exp(-bt))(2b+\sigma^{2})} \tag{10}\]
# Cybersecurity Threats and Their Mitigation Approaches Using Machine Learning--A Review
## 2 Background
### Cybersecurity Framework  
Cybersecurity forms an integral component of effective risk management in an organization. In order to handle cybersecurity risks, the role of NIST was modified to support the development of cybersecurity risk frameworks through the Cybersecurity Enhancement Act of 2014 [43]. This framework comprises three components, namely the framework core, implementation tiers and profiles. The framework core contains the basic guidance and standards that are imperative for an organization to manage risks posed by cyber threats. Implementation tiers proposed by NIST revolve around deciding on the scale of the proposed approach to mitigate threats. In other words, it allows an organization to understand the security standards required to guarantee protection. Finally, these frameworks support the creation of profiles that relate the cybersecurity activities to their respective outcomes. Profiles allow an organization to adapt their current approach to better suit demands.  
The NIST framework is primarily divided into five functions [44]. These are identify, protect, detect, respond and recover. Identify revolves around the organization's capability to understand and effectively manage risks posed to assets such as data and physical devices by cyber threats. Protect is responsible for ensuring security mechanisms for the safe transmission of vital data and resources. Detect ensures that the organization is ready to implement techniques that can effectively recognize cyber threats. Respond ensures  
Figure 3: Types of intrusion detection systems.  
that the organization is able to implement techniques that offer them the capability to respond to a threat. Finally, recover refers to activities that allow the organization to safely recover from a cybersecurity-related incident. Machine learning finds application in all these functions, especially in protect and detect. Protection categories such as access control can be implemented using machine learning. For example, NISTIR 8360 [45] utilizes a straightforward classification algorithm for verifying access control. Detect is perhaps the most widely explored area of machine learning. Almost all fields, such as anomaly detection and continuous monitoring, can benefit from a machine learning-based approached trained using a large amount of data. Discussion on machine learning techniques is given in the next section.
## 4 Test of the Anisotropic Damage Model  
This section evaluates the effectiveness of the proposed anisotropic damage model. A rectangular body, 25 mm \(\times\) 50 mm, subjected to an increasing downward displacement with confining pressure pc, was analyzed, as shown in Figure 5. The rectangle was divided into 12,806 triangular grids with a maximum length of 0.5 mm. The vertical displacement of the bottom boundary and the horizontal displacement of the lower left corner are fixed. The material parameters were considered as follows: \(E\) = 60 GPa, \(\
u\) = 0.3, \(\sigma_{\text{c0}}\) = 200 MPa, and \(\lambda_{\text{s}}\) = 0.1. The damage variable \(D\) was zero everywhere except within the crack band, where \(D\) was computed using Equation (12). Figure 6 shows the distribution of the damage variable and elastic modulus after failure. The proposed anisotropic damage model with two cases of confining pressure, \(p_{\text{c}}\) = 0 MPa and \(p_{\text{c}}\) = 10 MPa, was compared with the conventional isotropic damage model in the literature [44,45]. In the conventional isotropic damage model, the failure criterion is the Mohr-Coulomb strength criterion, and the shear damage evolution equation is based on compressive strain.  
Figure 6 shows the simulation results of failure modes corresponding to the two damage models. For the anisotropic damage model, only tangential slip occurs on the failure surface. For the isotropic damage model, the tangential slip and vertical compression occur on the failure surface. It is obviously unreasonable that the two fracture surfaces interpenetrate each other.  
Figure 5: Finite element model of the test case, (**a**) model geometry; (**b**) mesh grids.  
Figure 6: Comparison of failure modes (**a**) anisotropic damage model, \(p_{\text{c}}\) = 0 MPa; (**b**) anisotropic damage model, \(p_{\text{c}}\) = 10 MPa; (**c**) isotropic damage model, \(p_{\text{c}}\) = 0 MPa; (**d**) isotropic damage model, \(p_{\text{c}}\) = 10 MPa.