Delta Horizon Download [TOP] Setup Compressed

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Figure 12 shows the data block of the Compression Module. As shown, to perform delta encoding for each sensor, one data sample must be stored. Control of the sensor processed is made by a control module (Format Control). The stored data samples are subtracted from the new arrival samples. Delta-encoded data are compared with the range of symbols that will be compressed. A control module (Data Encoder Control) takes care of the codes to be Huffman encoded, the size of the data, the insertion of control codes, and to restore the data format with other control codes, such as timing information, card identification and sensor mask, and padding bits if needed. Huffman codes and their sizes are stored in 2 register banks and can be configured through commands. Although some latency is added to the compression process, the compression itself reduces the dead time due to the reduction of data to be sent through the available links.

Conceptually speaking, peridynamics incorporates some elements of molecular dynamics into a continuum mechanics framework. Based on continuum mechanics, consider a continuous body \({{\mathcal {B}}_0 \subset {\mathbb {R}}^3 }\) at time \(t=0\) in the material configuration and its spatial counterpart \({{\mathcal {B}}_t \subset {\mathbb {R}}^3}\), see Fig. 2. A point \({\mathbf {X}}\) in the material configuration is mapped to the spatial configuration by the nonlinear deformation map \({\mathbf {y}}\) as \({\mathbf {x}}={\mathbf {y}}({\mathbf {X}},t) : {\mathcal {B}}_0 \times {\mathbb {R}}_t \rightarrow {\mathcal {B}}_t\). Motivated by molecular dynamics, every point is influenced by the neighboring points within a finite distance. This region of interaction is called the peridynamic horizon \({\mathcal {H}}_0 \subset {\mathcal {B}}_0\) (Lagrangian perspective) with the horizon size \(\delta \) denoting the radius of the spherical neighborhood of \({\mathbf {X}}\) in the material configuration. In bond-based PD, the interaction between a point \({\mathbf {X}}\) and its neighbor \(\mathbf {X'}\) is considered via the undeformed bond vector \(\varvec{\Xi } = {\mathbf {X}}' - {\mathbf {X}}\) in the material configuration and the deformed bond vector \(\varvec{\upxi } = {\mathbf {x}}' - {\mathbf {x}} = {\mathbf {y}}({\mathbf {X}}') - {\mathbf {y}}({\mathbf {X}})\) in the spatial configuration.

Illustration of a continuum body \({\mathcal {B}}_0\) in the material configuration (left) and its spatial counterpart \({\mathcal {B}}_t\) (right). In PD, a point \({\mathbf {X}}\) interacts with its neighbors in a finite neighborhood \({\mathcal {H}}_0\) defined by the horizon size \(\delta \)

We consider the following setup. A thin stiff film is attached to a deep elastic substrate. The film is subject to a gradually increasing compression in horizontal direction. Once the effective strain in the film reaches a critical value \({\bar{\varepsilon }}_{crit}\), the flat surface loses stability and buckles to relax the compressive stresses. Since the film is not freestanding but bonded to the substrate, it is hindered from buckling into a single wave. Instead, short-wavelength buckling is the energetically favored equilibrium state of the system, which appears in form of sinusoidal surface wrinkles. In this paper, two different mechanisms inducing the compression state in the film are considered, which are schematically depicted in Fig. 4. The first mechanism is based on a substrate prestretch prior to film attachment. When the prestretch is released, the substrate relaxes to its initial length. Hence, the substrate is under tension while the film experiences compression triggering the instability. For the second mechanism, the critical conditions originate from a whole-domain compression. As the film is applied to an unstretched substrate, both components are in a state of compression at the onset of instability.

We study the influence of the stiffness ratio \(C_r = C_f/C_s\), with the elastic coefficients \(C_f\) and \(C_s\) of the film and the substrate, respectively, by assuming different values between 50 and 1000 at a constant film thickness of \(t=0.01\) and a constant horizon size of \(\delta = 0.0301\). The resulting wrinkling instabilities are analyzed in terms of their main features, i.e. the critical effective strain in the film \({\bar{\varepsilon }}_{crit}\) and the critical wavelength \(\lambda _{crit}\).

To study the influence of the horizon size \(\delta \) on the wrinkling characteristics, we carry out a nonlocality study. This is achieved through varying \(\delta \) while fixing the number of neighbors within the horizon by adjusting L according to \(\delta /L = 3.01\). In this manner, the observed effects can be attributed to the changing horizon sizes. With this study we pursue two objectives. The first goal is to analyze how the level of nonlocality of the material model affects the critical effective strain. Second, we aim at verifying our model through a comparison with the analytical solution for the classical model. It is known from the literature that the PD theory converges to the local solution of CCM for \({\delta \rightarrow 0}\). Therefore, we expect the numerical results to approach the analytical solution with decreasing \(\delta \). The numerical study is conducted with four different decreasing values for \(\delta \) (0.0602, 0.0301, 0.01505, 0.007525) at a constant film thickness \(t=0.02\). We test three different stiffness ratios \(C_r (50, 100, 200)\). We note that we are restricted in the choice of L and therefore also \(\delta \). Since we use a uniform grid and keep the film thickness t constant, it must be possible to resolve t by L.

Figure 7 illustrates the critical effective strain \({\bar{\varepsilon }}_{crit}\) as a function of the horizon size \(\delta \). For every modeled stiffness ratio the critical effective strain increases with increasing horizon size. It can be concluded that larger horizons, i.e. more nonlocal models, require larger effective strains to trigger the instabilities. Moreover, diminishing the horizon size leads to smaller differences between the nonlocal numerical results and the local analytical predictions. As expected, in the limit of \(\delta \rightarrow 0 \) the PD results approach the local solution. This confirms that our simulation model is capable of quantitatively predicting the critical conditions for surface wrinkles.

In contrast to a standard mesh convergence study for FEM, there are two types of convergence studies in PD due to the two independent parameters of grid spacing L and horizon size \(\delta \). One option is to fix \(\delta \) while decreasing L resulting in more neighbors within the horizon of each point. The discretized nonlocal solution is expected to converge to the continuum nonlocal solution. The second option is to keep a sufficient number of neighbors constant by fixing \(\delta /L\) while decreasing \(\delta \). In the limit of \({\delta \rightarrow 0}\) the nonlocal solution is expected to converge to the local solution. In this paper, we have conducted the second option in order to investigate the effect of nonlocality. In Fig. 7, the convergence of our PD solution to the local solution is shown. A comprehensive convergence study also following the first option has been carried out in our previous study [59], demonstrating that both types of convergence are achieved. \(\square \)

The next parameter of interest is the film thickness t. To investigate its influence on the critical effective strain \({\bar{\varepsilon }}_{crit}\), we consider four different values of t (0.01, 0.02, 0.03, 0.04). The study is conducted for six different stiffness ratios (50, 75, 100, 125, 150, 200) with a fixed horizon size \(\delta = 0.0301\). As is shown in Fig. 9, the numerical analysis predicts a decrease of critical effective strain with increasing film thickness. It is interesting to note that this relation is not covered by the analytical solution within the local theory which renders a constant critical effective strain independent of the film thickness. This observation suggests that the influence of t on \({\bar{\varepsilon }}_{crit}\) is introduced by nonlocality.

To further clarify the relation between the critical strain and the film thickness, we carry out a nonlocality study for three different film thicknesses (0.01, 0.02, 0.04). That is, we test four different horizon sizes (0.0602, \(0.0301, 0.01505, 0.007525)\) at a constant stiffness ratio \(C_r = 100\). We remark that since we adjust L according to \(\delta /L=3.01\) and use uniform grids, we are limited to a maximum horizon size \(\delta =0.0301\) for \(t=0.01\). Figure 10 compares the critical effective strain obtained numerically to the local analytical solution. It can be observed that the differences between the numerical results for different film thicknesses decrease with decreasing \(\delta \) and approach the local analytical result. Consequently, the data demonstrate that the influence of the film thickness on the critical effective strain can be considered a nonlocal effect. We again point out that the film thickness is the only length scale parameter inherent to the local theory. Since PD additionally includes the horizon size, an interplay of the two length scale parameters might cause the observed effect.

Figure 11 summarizes the results for the film thickness \(t=0.02\), horizon size \(\delta =0.0301\) and three different stiffness ratios \(C_r\) (100, 200, 500). The left graph shows an increase in critical wavelength with increasing \(\alpha \). Since a higher stiffness of the interfacial bonds leads to a higher overall stiffness of the interface layers, this trend is in line with the conclusion in Sect. 4.1.1. The results of the local analytical solution are included in the graph for the sake of completeness. It can be observed that the deviations of the numerical to the analytical results are larger for smaller values of \(\alpha \). However, we point out that the deviations cannot be explained solely by the variation of the interfacial stiffness as they might also result from nonlocality as well as the dependence of the wavelength on the computational domain as described in Sect. 4.1.1. Analyzing the right graph in Fig. 11, we make three observations. First, in the proximity of \(\alpha =0\) increasing \(\alpha \) causes a sharp increase of the critical effective strain. Second, once a maximum is reached, the critical effective strain continuously decreases. The latter effect can again be attributed to the higher overall stiffness of the interface region. However, the stiffness of the interface bonds also affects the strength of the attachment between the film and the substrate, resulting in the first observation. For \(\alpha \) approaching zero, the interface bonds are too weak to fully couple the film and the substrate. Consequently, less effective strain is required for the film to overcome the resistance of the substrate and buckle. Figure 12 further illustrates this effect by means of deformation plots for different values of \(\alpha \). Third, the local analytical solution consistently predicts smaller values than the numerical results. This agrees with the findings from Sect. 4.1.2, where we established that a nonlocal material model yields a higher critical effective strain than a local model. 2b1af7f3a8