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A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity.
The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly.
The computational complexity of the resulting method is proportional to the number of inclusions and this is optimal in comparison to data complexity.
The discretization error is proportional to the particle distance and does not depend on the conductivity contrast in the medium.
This paper presents some weighted H2-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., highly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of thermal conductivity of the material as the volume fraction of the particles is close to the jammed state. We proof that the characteristic percolation behavior of the material is well captured by standard conforming finite element models.
A Composite Finite Element Method approximates linear elliptic boundary value problems of Dirichlet type with discontinuous coefficients at possibly high contrast. The challenge is the discontinuity in the coefficient across some interface which is not necessarily resolved by the underlying finite element mesh. The method is non-conforming in the sense that shape functions preserve continuity across the interface only in an approximative way. However, the method allows to balance the non-conformity and the best approximation error in such a way that the total discretization error is optimal with regard to the mesh size and independent of contrast.