Darcy's Law for Flow in Porous Media and the Two-Space Method,

Keller, Joseph B.

doi:10.21236/ada095629

Cited by 46 publications

(47 citation statements)

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“…The multiple-scales technique that we exploit has been widely employed in homogenization of flow and transport in porous media in Keller (1980), Mei and Auriault (1991), Rubinstein (1987), and Tartar (1980), and more recently to simple models of growing tissue in O'Dea et al (2014) and Penta et al (2014) -here we seek to extend these ideas to a more complex description of the underlying tissue dynamics, incorporating multiple phases as well as microstructural features. We obtain a tissue-scale description of tumour growth and response, and transport of drug and nutrient comprising a system of advection-reaction PDEs that are partially parameterized by the solution to a tensor Stokes problem on a microscopic periodic unit.…”

Section: Discussionmentioning

confidence: 99%

“…There is an extensive literature related to the classical homogenization of flow and transport in porous media in both physical and biological applications, see, e.g. Keller (1980), Mei and Auriault (1991), Rubinstein (1987), and Tartar (1980). The analysis we present here represents an extension of the traditional homogenization of flow and transport phenomena and the recent attempts to apply these ideas to growing material in O'Dea et al…”

Section: Model Descriptionmentioning

confidence: 99%

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A multi-scale analysis of drug transport and response for a multi-phase tumour model

2016

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In this article we consider the spatial homogenization of a multiphase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations we employ an asymptotic homogenization of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction-advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug-and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

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Section: Discussionmentioning

confidence: 99%

Section: Model Descriptionmentioning

confidence: 99%

A multi-scale analysis of drug transport and response for a multi-phase tumour model

2016

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“…Since Theorem 0.1 proves the existence of the first term in the ansatz (0.3), the two-scale convergence method appears as the mathematically rigorous version of the, intuitive and formal, two-scale asymptotic expansion method [7], [10], [27], [40]. The key of the success for such a method is to consider only periodic homogenization problems.…”

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confidence: 99%

Homogenization and Two-Scale Convergence

Allaire¹

1992

SIAM J. Math. Anal.

1,929

2,637

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Abstract. Following an idea of G. Nguetseng, the author defines a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its "two-scale" limit, up to a strongly convergent remainder in L2 (12)) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar [10], [40] for details). This. method is very simple and powerful, but unfortunately is formal since, a priori, the ansatz (0.3) does not hold true. Thus, the two-scale asymptotic expansion method is used only to guess the form of the homogenized operator L, and other arguments are needed to prove the convergence of the sequence u to u. To this end, the more general and powerful method is the so-called energy method of Tartar [42]. Loosely speaking, it amounts to multiplying equation (0.1) by special test functions (built with the solutions of the cell equation), and passing to the limit as e-0. Although products of weakly convergent sequences are involved, we can actually pass to the limit thanks to some "compensated compactness" phenomenon due to the particular choice of test functions.Despite its frequent success in the homogenization of many different types of equations, this way of proceeding is not entirely satisfactory. It involves two different steps, the formal derivation of the cell and homogenized equation, and the energy method, which have very little in common. In some cases, it is difficult to work out the energy method (the construction of adequate test functions could be especially tricky). The energy method does not take full advantage of the periodic structure of the problem (in particular, it uses very little information gained with the two-scale asymptotic expansion). The latter point is not surprising since the energy method was not conceived by Tartar for periodic problems, but rather in the more general (and more difficult) context of H-convergence. Thus, there is room for a more efficient homogenization method, dedicated to partial differential equations with periodically oscillating coefficients. The purpose of the present paper is to provide such a method that we call two-scale convergence method.This new method relies on the following theorem, which was first proved by Nguetseng [36].

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“…[41]). The techniques required to perform this homogenisation are considerably more complex that those described for the interdigitated device in §4.1 and involve application of the formal asymptotic method of multiple-scales as originally developed by Joseph Keller [30,31] (relevant extensions of this method to applications in which the microstructure is not entirely periodic are given in [6,11,48,49]). We note further that there is an extensive literature on rigorous homogenization methods (see [1,14]) but that these methods involve considerably more effort than their formal counterparts whilst yielding exactly the same homogenized equations (they are also particularly hard to apply to microstructures that are not strictly periodic).…”

Section: A Bulk Heterojunction With Three-dimensional Microstructurementioning

confidence: 99%

Derivation and solution of effective medium equations for bulk heterojunction organic solar cells

2017

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A drift-diffusion model for charge transport in an organic bulk-heterojunction solar cell, formed by conjoined acceptor and donor materials sandwiched between two electrodes, is formulated. The model accounts for (i) bulk photogeneration of excitons, (ii) exciton drift and recombination, (iii) exciton dissociation (into polarons) on the acceptor-donor interface, (iv) polaron recombination, (v) polaron dissociation into a free electron (in the acceptor) and a hole (in the donor), (vi) electron/hole transport and (vii) electron-hole recombination on the acceptor-donor interface. A finite element method is employed to solve the model in a cell with a highly convoluted acceptor/donor interface. The solutions show that, with physically realistic parameters, and in the power generating regime, the solution varies little on the scale of the microstructure. This motivates us to homogenise over the microstructure; a process that yields a far simpler one-dimensional effective medium model on the cell scale. The comparison between the solution of the full model and the effective medium (homogenised) model is very favourable for the applied voltages that are less than the built-in voltage (the power generating regime) but breaks down as the applied voltages increases above it. Furthermore, it is noted that the homogenisation technique provides a systematic way to relate effective medium modelling of bulk heterojunctions [19,25,36,37,42,59] to a more fundamental approach that explicitly models the full microstructure [8,38,39,58] and that it allows the parameters in the effective medium model to be derived in terms of the geometry of the microstructure. Finally, the effective medium model is used to investigate the effects of modifying the microstructure geometry, of a device with an interdigitated acceptor/donor interface, on its current-voltage curve.

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Darcy's Law for Flow in Porous Media and the Two-Space Method,

Cited by 46 publications

References 0 publications

A multi-scale analysis of drug transport and response for a multi-phase tumour model

A multi-scale analysis of drug transport and response for a multi-phase tumour model

Homogenization and Two-Scale Convergence

Derivation and solution of effective medium equations for bulk heterojunction organic solar cells

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