Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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8 MATHEMATICAL AND NUMERICAL METHODS
8 MATHEMATICAL AND NUMERICAL METHODS
Pages 88-96
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From page 88... ...
On the macroscopic level, materials properties are generally represented by a set of partial differential equations that express energy, mass, and momentum conservation and are formulated to represent the symmetry of the material to which they are applied. On the mesoscopic level, theoretical approaches are less well defined.
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From page 89... ...
The use of parallel computers for electronic structure calculations has been developing slowly because of the size of the computational codes, uncertainty in possible computational gains, and the difficulty in adjusting well-developed procedures to novel computational environments. Most parallel computing in this area has involved the joint efforts of a materials theorist, an applied mathematician, and a computer scientist.
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From page 90... ...
In quantum mechanical problems, where the above mentioned density functional theory is used, the Kac-Feynman path integral is also often used as the basis for analytic and numerical work (Feynman, 19723. In classical statistical mechanics, analogous mathematical constructs called Wiener integrals and density functionals exist.
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From page 91... ...
for fracture prediction based on simple crack models (Babuska and Miller, 1984; Vasilopoulos, 1988; Sumaratna and Ting, 1986; Reitich, 1991~. Surface-stability analysis of moving interfaces has led to an increased understanding of microstructural growth during solidification; see two subsections, Phase Transformations and Pattern Formation and Dendritic Growth, in Chapter 4.
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From page 92... ...
Much work on the theory and numerical solution of partial differential equations under these challenging conditions awaits attention. Often in materials research, the scientist is interested in inferring from measurements made on a macroscopic level information about the mesoscale.
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From page 93... ...
As a material interface or free-moving surface becomes unstable, rich complex structures evolve into columnar grains, lamelIar eutectics, dendritic growths of fractal structures, and other structural features. Both views of microstructural evolution, that of evolving phase transitions and that of instability development in interfaces and surfaces, point to the often exceptionally complex mesoscopic morphology of materials.
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From page 94... ...
~' Mere are also related approaches, expressing time-dependent position and momentum and that include viscous and random forcing terms, based on Langev~n equations. The theorist is now faced with solving sets of nonlinear dissipative stochastic partial differential equations that are first-order in time.
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From page 95... ...
Directly porting a serial or vector algorithm to a parallel computer typically leads to disappointing performance gains. Many of the equations, boundary conditions, geometries, initial conditions, and so on that are germane to materials problems differ from those traditionally addressed by applied mathematicians in such applications as hydrodynamics.
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From page 96... ...
This correspondence follows from analyses analogous to those used in statistical mechanics for determining macroscopic behavior in the kinetic theory of gases, from which the Navier-Stokes equations for a collection of interacting particles can be derived. For other physical systems, more analysis and understanding are needed.
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