• Corpus ID: 219966153

A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution

@article{Milton2020AUP,
  title={A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution},
  author={Graeme W. Milton},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
  • G. Milton
  • Published 15 June 2020
  • Physics, Mathematics
  • arXiv: Mathematical Physics
We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites… 

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References

SHOWING 1-10 OF 25 REFERENCES
A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint
We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an
A new route to finding bounds on the generalized spectrum of many physical operators
  • G. Milton
  • Physics, Mathematics
    Journal of Mathematical Physics
  • 2018
Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a
Convergence of iterative methods based on Neumann series for composite materials: theory and practice
Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential
Exact Relations for Effective Tensors of Polycrystals. I. Necessary Conditions
Abstract.The set of all effective moduli of a polycrystal usually has a nonempty interior. When it does not, we say that there is an exact relation for effective moduli. This can indeed happen as
A unifying perspective on linear continuum equations prevalent in physics. Part I: Canonical forms for static and quasistatic equations
Following some past advances, we reformulate a large class of linear continuum physical equations in the format of the extended abstract theory of composites so that we can apply this theory to
The Theory of Composites
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
Variational and Related Methods for the Overall Properties of Composites
Publisher Summary This chapter focuses on variational and related methods for the overall properties of composites. A wide range of phenomena that are observable macroscopically are governed by
Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions
Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that
On characterizing the set of possible effective tensors of composites: The variational method and the translation method
A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the
A polarization based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast
It is recognized that the convergence of FFT based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases.
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