Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics

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The Theory of Composites

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

/pdf/the-heterogeneous-multiscale-method-4zoj61bb2c.pdf

The heterogeneous multiscale method

Simulation of the multi-scale convergence in computational homogenization approaches

https://www.math.nyu.edu/faculty/majda/pdfFiles/PhysRep%201999%20Majda%20Kramer.pdf

Simplified models for turbulent diffusion : theory, numerical modelling, and physical phenomena

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.

V. V. Zhikov

Papers

A one dimensional variational problem with gradient constrained

Homogenization of Navier-Stokes systems for electrorheological fluids