A new boundary condition for homogenization of high-contrast random heterogeneous materials

Abstract

Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition.

Keywords:

• boundary condition
• high contrast
• effective coefficient
• condition number
• stochastic homogenization

2010 AMS Subject Classifications:

• 65N12
• 65N30
• 80M40

Disclosure

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [No. 11471262, No. 11071196].