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By Dahlquist G.

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Numerical Analysis and Its Applications: Second InternationalConference, NAA 2000 Rousse, Bulgaria, June 11–15, 2000 Revised Papers

This booklet constitutes the completely refereed post-proceedings of the second one overseas convention on Numerical research and Its purposes, NAA 2000, held in Rousse, Bulgaria in June 2000. The ninety revised papers provided have been rigorously chosen for inclusion within the publication in the course of the rounds of inspection and reviewing.

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Then, if one defines the concave polar of g by g∗ (x, q) = inf {sq − g(x, s)}, (24) g(x, s) ≤ inf {sq − g∗ (x, q)}, (25) s≥0 it follows that q≥0 with the equality holding true if g is concave. Assuming then that the complementary energy density function w ∗ of the nonlinear heterogeneous material is such that g is concave, it follows from (25) that w∗ (x, D) = inf {w 0∗ (x, D) + v(x, 0 0 ≥0 )}, (26) where q has been identified with (2 0 )−1 and s with D 2 , such that w 0∗ (x, D) = [ 12 0 (x)]D 2 is the complementary-energy function of the linear, heterogeneous comparison material with arbitrary non-negative dielectric coefficient 0 (x), and v(x, 0 ) = g ∗ (x, 12 0 ).

In the next two chapters we will also describe the macroscopic behavior of nonlinear materials that can be described by a few other types of nonlinear constitutive equations, for which considerable progress has been made, and a comparison between the theoretical predictions and the experimental data is possible. (2) In the second class of nonlinearities, a material is characterized by thresholds in the (local as well as macroscopic) potential gradient. Then, depending on the physics of the phenomenon under study, one of the following two scenarios may arise.

For example, it has been suggested that strong local field effects, such as the large local field at the surface plasmon resonance frequency of a metallic inclusion, may lead to enhanced nonlinear response in a heterogeneous material. Constitutive nonlinearity is the subject of this and the next two chapters. Even within this restricted class of nonlinear materials, one may imagine a very large number of nonlinear constitutive equations (similar to those that have been proposed, for example, for polymeric fluids).

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