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Multiscale numerical methods for the simulation of diffusion processes in random heterogeneous media with guaranteed accuracy

Alves Paladim, Daniel 2016. Multiscale numerical methods for the simulation of diffusion processes in random heterogeneous media with guaranteed accuracy. PhD Thesis, Cardiff University.
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Abstract

The possibility of combining several constituents to obtain properties that cannot be obtained with any of them alone, explains the growing proliferation of composites in mechanical structures. However, the modelling of such heterogeneous systems poses extreme challenges to computational mechanics. The direct simulation of the aforementioned gives rise to computational models that are extremely expensive if not impossible to solve. Through homogenisation, the excessive computational burden is eliminated by separating the two scales (the scale of the constituents and the scale of the structure). Nonetheless, the hypotheses under which homogenisation applies are usually violated. Traditional homogenisation schemes provide no means to quantify this error. The �rst contribution of this thesis is the development of a method to quantify the homogenisation error. In this method, the heterogeneous medium is represented by a stochastic partial di�erential equation where each realisation corresponds to a particle layout. This representation allows us to derive guaranteed error estimates with a low computational cost. The e�ectivity (ratio between true error and estimate) is characterised and a relation is established between the error estimates and classical results in micromechanics. Moreover, a strategy to reduce the homogenisation error is presented. The second contribution of this thesis is made by developing a numerical method with guaranteed error bounds that directly approximates the solution of heterogeneous models by using shape functions that incorporate information of the microscale. The construction of those shape functions resembles the methods of computational homogenisation where microscale boundary value problems are solved to obtain homogenised properties.

Item Type: Thesis (PhD)
Status: Unpublished
Schools: Engineering
Uncontrolled Keywords: Finite Element Method; Numerical Methods; Error Estimation; Homogenisation; Multiscale; Mechanics
Date of First Compliant Deposit: 9 May 2017
Last Modified: 04 Jun 2017 09:49
URI: http://orca.cf.ac.uk/id/eprint/100344

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