SLIDE 1 Error estimation in homogenisation Error estimation in homogenisation
Daniel Alves Paladim1
(alvesPaladimD@cardifg.ac.uk)
Pierre Kerfriden1 José Moitinho de Almeida2 Stéphane P . A. Bordas1,3
1School of Engineering, Cardifg University 2Instituto Superior T
écnico, Universidade de Lisboa
3Faculté des Sciences, Université du Luxembourg
Strobl, 27th of January, 2015
SLIDE 2
Motivation
Problem: Analysis of an heterogeneous materials. Vague information available. The position of the particles is not available. 2
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Motivation
Problem: Analysis of an heterogeneous materials. Vague information available. The position of the particles is not available. Solution: Homogenisation. 2
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Motivation
Problem: Analysis of an heterogeneous materials. Vague information available. The position of the particles is not available. Solution: Homogenisation. New problem: Assess the validity of the homogenisation. 2
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Proposed solution
Idea: Understand the original problem as an SPDE (the center of particles is a random variable) and bound the distance between both models 3
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Proposed solution
SPDE: Stochastic partial difgerential equation. Collection of parametric problems + probability density function 3
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QoI: Quantity of interest. The output. Scalar that depends of the solution.
Proposed solution
(linear) 3
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QoI: Quantity of interest. The output. Scalar that depends of the solution.
Proposed solution
(linear) 3
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Problem statement
Heterogeneous problem
Heat equation
4
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Problem statement
Heterogeneous problem Homogeneous problem
Heat equation
4
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Problem statement
Heterogeneous problem Homogeneous problem Aim: Bound The computation of the bound must be deterministic.
Heat equation
4
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Derivation
Hypothesis Deterministic boundary conditions 5
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Derivation
Hypothesis Deterministic boundary conditions Constant volume fraction 5
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Derivation
Hypothesis Deterministic boundary conditions Constant volume fraction Constant PDF over the domain 5
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“Flux” FE
The unknown is the fmux fjeld and are fulfjlled strongly. 6
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“Flux” FE
The unknown is the fmux fjeld and are fulfjlled strongly. In contrast, in “temperature” FE , the temperature is the unknown and 6
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“Flux” FE
The unknown is the fmux fjeld and are fulfjlled strongly. In contrast, in “temperature” FE , the temperature is the unknown and In order to derive bounds, we will also need to use an homogenised “fmux” FE solution 6
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Error in the energy norm
Rewriting the problem in terms of the fmux and the temperature 7
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Error in the energy norm
Rewriting the problem in terms of the fmux and the temperature will fulfjll exactly the fjrst 2 equations. 7
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Error in the energy norm
Rewriting the problem in terms of the fmux and the temperature will fulfjll exactly the fjrst 2 equations. will fulfjll exactly the 3rd equation. 7
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Error in the energy norm
Rewriting the problem in terms of the fmux and the temperature will fulfjll exactly the fjrst 2 equations. will fulfjll exactly the 3rd equation. In general, Discrepancy = measure of the error 7
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Error in the energy norm
Formalizing this idea, it can be shown that 8
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Error in the energy norm
Formalizing this idea, it can be shown that Expanding 8
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Error in the energy norm
Formalizing this idea, it can be shown that Deterministic quantity Expanding 8
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Error in the quantity of interest
The error in energy norm is not always relevant. Solution: Bound for the quantity of interest 9
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Error in the quantity of interest
The error in energy norm is not always relevant. Solution: Bound for the quantity of interest Dual problem 9
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Error in the quantity of interest
The error in energy norm is not always relevant. Solution: Bound for the quantity of interest Dual problem 9
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Error in the quantity of interest
Cauchy-Schwarz inequality The error in energy norm is not always relevant. Solution: Bound for the quantity of interest Dual problem 9
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Error in the quantity of interest
Cauchy-Schwarz inequality Use the bound in the energy norm, The error in energy norm is not always relevant. Solution: Bound for the quantity of interest Dual problem 9
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Validation
The “exact” quantity of interest is computed with 512 MC realisations. The quantity of the interest is the average temperature in the exterior faces. 10
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Validation
The “exact” quantity of interest is computed with 512 MC realisations. The quantity of the interest is the average temperature in the exterior faces. 10
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Validation
11
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Validation
Studied in a domain homogenised through rule of mixture. 12
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Validation
Studied in a domain homogenised through rule of mixture. Dual problem 12
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Validation
Studied in a domain homogenised through rule of mixture. Dual problem T wo problems solved twice: – Using “temperature” FE – Using “fmux” FE 12
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Validation
13
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Validation
Around 60 000 elements 512 problems, 512 difgerent meshes Full PDF Around 2000 elements 4 problems, 1 mesh Bounds on the expectation 14
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Work in progress
– A bound on the variance. 15
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Work in progress
– Enhanced model. Insert patches with particles in parts of the domain. 15
SLIDE 40
Summary
– A bound for the homogenisation error was presented. – The computation of the bound is deterministic. – The error estimate, should be used with care when there is a high contrast between the material properties. 16
SLIDE 41
Summary
– A bound for the homogenisation error was presented. – The computation of the bound is deterministic. – The error estimate, should be used with care when there is a high contrast between the material properties.
Thank you for your attention.
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