An adaptive scheme for homogenised domains

In this paper, we extend the concept of modelling error estimation for the homogenisation of ellipticPDEs. In order to do so, we fully acknowledge that the rapid spatial variation of microscopicdiffusion constants cannot be known exactly. Therefore, we represent the microscopic diffusioncoefficients as a random field. In this context, the accuracy of surrogate models, such ashomogenisation schemes, can be quantified by estimating the error in the first moments of theprobability density function of a quantity of interest.We propose a way to bound the error in the two first moments, following and extending the seminalwork of [1]. Our derivations rely on the Constitutive Relation Error[2] (CRE), which states that acertain distance between the solutions delivered by the primal and a dual surrogates of the originalstochastic problem is equal to some measure of the exact and unaffordable errors. We further assumethat these surrogates are deterministic, consistently with the theory of homogenisation. Minimisingthe CRE in this subset of homogenisation schemes leads us to an optimal surrogate that is closelyrelated to the classical Voigt and Reuss models. This result is used in a goal-oriented setting toestablish upper and lower bounds for the first two moments of the quantity of interest. We show that the method respect the numerical separation of scales, and is therefore affordable andeasy to implement, and that it produces useful results as long as the mismatch between the diffusioncoefficients of the microstructure remains small. We will propose extensions for the case of highmismatch, by allowing the surrogate solutions to fluctuate in the stochastic domain

Plasma rico em plaquetas combinado a hidroxiapatita na formação do calo ósseo em fraturas induzidas experimentalmente no rádio de cães Platelet-rich plasma combined with hydroxyapatite on bone callus formation in experimental radii fractures in dogs

Avaliou-se, no presente estudo, os efeitos do plasma rico em plaquetas (PRP) autógeno combinado a hidroxiapatita não-absorvível na formação do calo ósseo em fraturas diafisárias induzidas experimentalmente em rádio de cães e fixadas com placas e parafusos ósseos. Foram utilizados oito cães adultos sem raça definida, machos ou fêmeas, com pesos compreendidos entre 5 e 15kg. Induziu-se na diáfise de ambos os rádios defeito ósseo correspondente a 25% do diâmetro do mesmo. No rádio direito (grupo HA/PRP), implantou-se plasma rico em plaquetas combinado a hidroxiapatita. No rádio esquerdo (grupo HA), implantou-se somente a hidroxiapatita. Para a obtenção do PRP, empregou-se protocolo simplificado utilizando centrífuga laboratorial comum. Comparou-se entre os grupos a evolução clínica dos animais, o tempo de formação e a densidade do calo ósseo por exames radiográficos simples e pela técnica de densitometria óptica aos 15, 30, 45 e 60 dias de pós-operatório. Tanto na avaliação radiográfica como densitométrica não se observou diferença estatisticamente significante na evolução do calo ósseo entre os grupos (P>0,05). No presente estudo, o PRP autógeno combinado à hidroxiapatita não-absorvível não acelerou o processo de formação do calo ósseo, comparado à hidroxiapatita somente.<br>The present study was aimed at evaluating the effects of autologous platelet-rich plasma (PRP) in combination with unabsorbable hydroxyapatite on bone callus formation in an experimental canine diaphyseal radius fracture stabilized with bone plate. Eight mature, male and female mongrel dogs were selected weighed 5 to 15kg. Bone defects were created, bilateral, in both radii diaphysis, corresponding to 25% of bone diameter. On the right radius (group HA/PRP), the defect was filled with PRP and hydroxyapatite. On the left radius (group HA), was placed hydroxyapatite alone. PRP was obtain by a simplified technique using a laboratory centrifuge. Clinical evaluations, bone formation and bone density by use of radiographic and optical densitometry, were compared between group HA/PRP and group HA after 15, 30, 45 and 60 days post surgery. The radiographic and densitometric study did not indicate a significant difference in the bone callus formation between the groups (P>0.05). In the present study the autologous PRP in combination with unabsorbable hydroxyapatite could not accelerated the bone callus in comparison with hydroxyapatite alone