Shapes Optimized

Banichuk N.V. (1980) Shape optimization of elastic bodies. Nauka, Moscow (in Russian). 

Haslinger J., Neittaanmaki P., Tiihonen T. (1986) Shape optimization in contact problems based on penalization of the state inequality. Apl. Mat. 31 (1), 54-77. 

Sokolowski J., Zolesio J.-P. (1992) Introduction to shape optimization. Shape sensitivity analysis. Springer-Verlag. 

In a computational sense, the problem is converted into a multidimensional discrete shape optimization problem in a space spanned by the various orderings of... 

J. C. McClure, Tool Wear and Shape Optimization in the Friction-Stir Welding of Aluminum Metal-Matrix Composite, Friction Stir Welding and Processing II,... 

By a shape optimization procedure that uses results from finite element or another analysis. 

The ultimate goal of MOR is to obtain reduced models for fast and efficient simulation, design, and control of microfluidic devices. In this section, we consider key research findings of MOR including fluid flow, species transport, liquid filling, droplet traffic, and shape optimization that have proven useful for system-level simulation and design and dynamic control. 

Shape optimization of microfluidic structures is a challenging problem, where MOR is strongly desired to reduce the computational complexity during iterations. Utilization of reduced order models for shape optimization in microfluidic devices has been explored recently. Antil et al. [15] combined the POD and the balanced truncation MOR methods for shape optimization of capillary barriers in a network of microchannels. Ooi [9] developed a computationally efficient SVM surrogate model for optimization of a bioMEM microfluidic weir to enhance particle trapping. 

Antil H, Heinkenschloss M, Hoppe RHW, Linsenmann C, Wxiforth A (2012) Reduced order modeling based shape optimization of surface acoustic wave driven microfluidic biochips. Math Comput Simul 82(10) 1986-2003... 

The fact that water-methanol shows a 60% higher viscosity maximum makes it much less attractive than water - acetonitrile from a purely kinetic standpoint. The extremely high maxima of the longer chain alcohols virtually exclude them from standard application use (let alone the tax issue with ethanol). Nevertheless, the positive impact of the different solvation properties of the alcohoi modifiers should never be underestimated for selectivity and peak shape optimization. 

J. A. Bennett and M. E. Botkin Shape Optimization Of Two-Dimensional Structures With Geometric Problem Description and Adaptive Mesh Refinement, General Motors Research Laboratories, May 2,1983. 

Fourie, PC. Groenwold, A.A. 2002. The particle swarm optimization algorithm in size and shape optimization. Structural and Multidisciplinary Optimization 23, 259-267. 

As it is seen, the values (Vc) and (Vc) are rather close, so the mode of metastable nucleation is difficult to reveal. Moreover, as discussed later, shape optimization excludes this mode (if we neglect stresses). 

It is evident that, since concentration gradient suppresses nuclei growth along the longitudinal direction, nature will find the possibility to increase the nucleus s volume (and reduce Gibbs free energy) by transversal growth. Hiis statement means that the nuclei formed in the diffusion zone must be nonspherical. Because of this, it is necessary to take into account the shape optimization for each fixed volume of a nucleus. 

Figure 4.6 Dependence of the Gibbs free energy on the volume at shape optimization (a) Vc > (b)Vc <...

Further, the dependence AG(V) is computed under the condition of shape optimization for each value of volume. Introduction of the elastic energy changes the general form of the dependence only quantitatively. The main result of the previous theory remains the same there exists a certain critical gradient above which nucleation is forbidden. 

Therefore, the transversal mode with shape optimization gives us the same qualitative results as the polymorphic mode ... 

A numerical analysis of Equation 5.4 with Ag(c) in the form of Equation 5.6 was done for different widths of the diffusion zone down to fC = 0.5 nm. We considered two different cases (i) without shape optimization (nucleus being a cube 21 X 21 X 21 and the shape parameter is fixed attp = ) and (ii) with shape optimization, that is, for each nucleus volume tp is determined to minimize AG. The calculated results demonstrate, in both cases, that even at a very narrow interdiffusion width the nucleation barrier is rather low, about 20kBT (Figure 5.4), and the critical size 21 o. of the nucleus amounts to 0.45 nm, which nearly coincides with the value of classical theory and implies practically immediate nucleation. 

Figure 5.4 Nucleation barrier of AI9C02 nucleus formation at the Al-Co interface calculated by the transversal mode without shape optimization for different diffusion zone widths.

The polymorphic mode of nucleation is characterized by the conservation of the concentration profile, which is a reasonable assumption if the system has no time to redistribute atoms inside and outside the nucleus in the process of lattice transformation. For the polymorphic mode, we also consider the two cases discussed earlier (i) without shape optimization (rp = 1) and (ii) with shape optimization. 

Figure S.6 Section through the AC(N,Cx) surfaces for the polymorphic mode without shape optimization at a composition Cy = 15 at.% (a = 8 X 10 J/atom, calculation for different diffusion zone widths /f) = 3 nm K2 = 3.5 nm ...

If the system has the kinetic opportunity of shape optimization, the resulting nucleation barrier decreases, as expected. In the case of full optimization, the surface AG (u, Cx) does not reveal a minimum. After passing the saddle point, it is always favorable to increase the volume by transversal growth, but keeping limited the longitudinal size along the direction of the gradient. A critical diffusion zone width, close to the experimental value, is obtained at a = 4 x 10 J/atom. This result implies that, for this parameter a and the diffusion zone width K = 3.5 nm. 

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