| Summary: | 碩士 === 國立交通大學 === 應用數學系所 === 98 === In the most general sense, a heterogeneous material is one that is composed of
domains of different materials (or phases), such as a composite, or the same
material in different states, such as a polycrystal. In many instances, the mi-
crostructures can be characterized only statistically, and therefore are referred
to as random heterogeneous materials(or random media), the chief of this study.
Consider an elliptic equation :
−div(A(ε−1 x, ω)∇uε (x, ω)) = f (x) on Q,
uε (x, ω)| = 0 on ∂Q;
where A, f, and u are in suitable function spaces , ω ∈ Ω and (Ω, Σ, μ) is a
suitable probability space. In this study we introduce the ergodic dynamical
systems on the probability space to describe the random media; we show the
matrix A(x, ω) above admits homogenization( see Definition.4.2) and the ho-
mogenized matrix is independent of ω ∈ Ω.
We give definitions, examples, and proofs about ergodic dynamical systems
in section two. Section three is about definition of realizations, and the ergodic
theorem. In section four, we recall the definition of homogenization of ellip-
tic equations for individual cases and statistical cases, and use the auxiliary
equations to define the homogenized matrix, and prove the main convergence
theorem through the div-curl lemma. In section five, we define the random
sets of the percolation, consider the existence of the effective conductivity, and
state the theorem of the existence of the effective conductivity of such random
media.
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