Cuboid Domain Research Articles

Numerical and analytical estimates of existence regions for semi-linear elliptic equations with critical Sobolev exponents in cuboid and cylindrical domains

We use a variety of careful numerical and semi-analytical methods to investigate two outstanding conjectures on the solutions of the parametrised semi-linear elliptic equation Δu+λu+u 5=0, u>0 , where u is defined to be zero on the boundary of a three dimensional domain. This equation is important in analysis and in studies of combustion and polytropic gases. It is known that there is a value λ 0>0 such that no solutions exist for λ< λ 0. McLeod and Schoen, and Bandle and Flucher have given different estimates for λ 0; both of which have been conjectured to be exact. We perform a semi-analytic study of solutions on cylindrical domains and construct numerical approximations on cuboid domains using the finite element method combined with a careful post-processing step to reduce the otherwise significant errors. We compute these estimates for cylindrical and cuboid domains, and show that on these domains the estimates do not agree. We conclude that the conjecture of Bandle and Flucher is false. Our numerical computations on cuboid domains are consistent with McLeod's conjecture being true.

Satisfaction of boundary conditions for Chebyshev collocation methods in cuboidal domains

A collocation strategy for the satisfaction of boundary conditions in the application of Chebyshev spectral methods to Poisson and biharmonic-type problems in cuboidal domains is presented. This strategy leads, in both cases, to nonsingular linear systems for the determination of the unknown coefficients.

Structure of a rapidly solidified Mn-16.2 at.% Al-12 at.% C alloy

The structure of a rapidly solidified Mn-16.2 at.% Al-12 at.% C alloy has been investigated by means of transmission electron microscopy. The rapidly solidified (melt-spun) alloy contained an fcc γ-Mn phase and an ordered fcc phase. The ordered phase appeared to be as cuboidal domains wetted by the disordered fcc γ-Mn phase. Structural analysis by electron diffraction suggested that the crystal structure of the ordered fcc phase formed in the rapidly solidified Mn-16.2 at.% Al-12 at.% C alloy is an interstitially ordered Ll 2, i.e. L l 2 perovskite-type structure.

Electron diffraction structure analysis of rapidly solidified (Fe,Ni) 3Al-C alloys

The quench-rate-metastable structure was examined in the Fe-20Ni-8Al-2C (weight percent) alloy. Duplex structures, either disordered γ FCC with κ-carbide precipitates or cuboidal domains of ordered γ' with the L1 2 structure in a disordered γ matrix, were found in samples 35–100 μm thick. Electron diffraction patterns, from a series of samples with different quench rates, were analyzed with image analysis techniques to trace the development of the ordered γ' phase.

On the Stress Field and Surface Deformation in a Half Space With a Cuboidal Zone in Which Initial Strains Are Uniform

A solution is given for the stress field and surface deformation in a half space in the presence of uniform inelastic strains in a subsurface cuboidal domain. The problem is solved by using the full space solution of two cuboids with specified initial strain and relieving the stress acting normal to their plane of symmetry. The solution for stresses and surface deformations are obtained in closed form. Numerical results are given for stresses and surface displacement as a function of the depth of the cuboidal element.