Limit of a Consistent Approximation to the Complete Compressible Euler System

The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of the compressible complete Euler system in full space $$ \mathbb {R}^d,\; d=2,3 $$ R d , d = 2 , 3 is a weak solution of the system, then the approximate solutions eventually converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.

Historical Tools of Anthropometric Facial Assessment: A Systematic Raw Data Analysis on the Applicability of the Neoclassical Canons and Golden Ratio

Background The fundamental tenets of facial aesthetic surgery education have not changed in centuries. Research is beginning to demonstrate that the Neoclassical Canons and the Golden Ratio, Phi, have limited use in populations other than those of White European extraction. Objectives The purpose of this study is to analyze the comparable raw data in the literature to determine 1) if there is interethnic variability in Neoclassical Canon and Phi measurements and 2) if the measurements in these representative samples differ from the “ideal.” Methods A PubMed/Scopus search was performed. Manuscripts with raw data and individuals aged ≥16 were included. Measurements were extracted and used to calculate the Neoclassical Canons and Phi. One-way ANOVA tests were run to compare mean measurements across six ethnic groups. p<0.05. Results Twenty-seven articles were included. Every continent was represented except Antarctica and Australia. Men were less commonly studied than women. Subject ages ranged from 16 to 56. Averaged Canons 2, 6-8 measurements had significant interethnic differences in males whereas Canons 5-8 had significant differences across ethnicities in females. For men, there was significant interethnic variability in measurements of Phi 2, 5, 8, 10 and 17. For women, Phi 1, 2, 5, 8, 10 and 17 varied across ethnicities. No ethnic/gender group showed consistent approximation of the “ideal” for both the Neoclassical Canons and Phi. Conclusions Today, the utility of the Neoclassical Canons and Phi is limited. It is incumbent on our field to systematically study and define the anthropometric measures that define the “ideal.”

Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach

Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays

Wave propagation and run-up in U-shaped channel bays are studied here in the framework of the quasi-1D Saint-Venant equations. Our approach is numerical, using the momentum conserving staggered-grid (MCS) scheme, as a consistent approximation of the Saint-Venant equations. We carried out simulations regarding wave focusing and run-ups in U-shaped bays. We obtained good agreement with the existing analytical results on several aspects: the moving shoreline, wave shoaling, and run-up heights. Our findings also confirm that the run-up height is significantly higher in the parabolic bay than on a plane beach. This assessment shows the merit of the MCS scheme in describing wave focusing and run-up in U-shaped bays. Moreover, the MCS scheme is also efficient because it is based on the quasi-1D Saint-Venant equations.

Consistent Approximation of Fractional Order Operators

Fractional order controllers become increasingly popular due to their versatility and superiority in various performance. However, the bottleneck in deploying these tools in practice is related to their implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is a foundation. Generally, the following three identical equations always hold, i.e., $\frac{1}{s^\alpha}\frac{1}{s^{1-\alpha}} = \frac{1}{s}$, $s^\alpha \frac{1}{s^\alpha} = 1$ and $s^\alpha s^{1-\alpha} = s$. However, for the approximate models of fractional differintegrator $s^\alpha$, $\alpha\in(-1,0)\cup(0,1)$, there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacious in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.