Journal
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Volume 20, Issue 7, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0965-0393/20/7/075008
Keywords
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Funding
- National Science Foundation [0511232, 0502737, 0854905, 0542619, 0907030, 0746424]
- Direct For Mathematical & Physical Scien [746424, 0542619, 0854905] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Materials Research [0907030] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0511232] Funding Source: National Science Foundation
- Division Of Materials Research [746424, 0542619] Funding Source: National Science Foundation
- Division Of Materials Research
- Direct For Mathematical & Physical Scien [0502737] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [0854905] Funding Source: National Science Foundation
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In this paper, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against numerical solutions of the corresponding sharp-interface equations. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces and relaxation of a droplet on irregular surfaces.
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