Advances in error estimation for Advances in error estimation for homogenisation homogenisation

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Advances in error estimation for

homogenisation

Daniel Alves Paladim1 (alvesPaladimD@cardiff.ac.uk)

Pierre Kerfriden1

José Moitinho de Almeida2

Stéphane P. A. Bordas1,3 1_{School of Engineering, Cardiff University}

2_{Instituto Superior Técnico, Universidade de Lisboa} 3_{Faculté des Sciences, Université du Luxembourg}

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Motivation

Problem: Analysis of an heterogeneous materials. Vague

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Motivation

information available. The position of the particles is not available.

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Motivation

information available. The position of the particles is not available.

Solution: Homogenisation.

New problem:

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Key ideas

Exact model

● To estimate error, we need a reference to compare our solution ● Reference: solution of an stochastic PDE

● Able to take into account the vague description of the domain

Error estimation

● Objective: Compare the solution of the two models (without solving

the SPDE)

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Proposed solution

Idea: Understand the original problem as an SPDE (the center of particles

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Proposed solution

SPDE: Stochastic partial differential equation.

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QoI: Quantity of interest. The output. Scalar that depends of the solution.

Proposed solution

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QoI: Quantity of interest. The output. Scalar that depends of the solution.

Proposed solution

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Problem statement

Heterogeneous problem

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Problem statement

Heterogeneous problem Homogeneous problem

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Problem statement

Heterogeneous problem Homogeneous problem

Aim: Bound

The computation of the bound must be deterministic.

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Hypothesis

Hypothesis

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Hypothesis

Hypothesis

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Hypothesis

Hypothesis

Deterministic boundary conditions

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Hypothesis

Hypothesis

Deterministic boundary conditions

Knowledge of the probability of being inside particle for every point of the domain.

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Outline

Error estimation

● Objective: Compare the solution of the two models (without

solving the SPDE)

● To estimate the error, an equilibrated flux field is needed

● With an equilibrated flux field, we can estimate the error in

energy norm

● And with an estimator for the error in energy norm, we can

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Equilibrated flux field

An equilibrated flux field fulfills

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Equilibrated flux field

strongly.

In contrast, in “temperature” FE , the temperature is the unknown and

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Equilibrated flux field

strongly.

In contrast, in “temperature” FE , the temperature is the unknown and

In order to derive bounds, we will use flux FE to compute an homogenised equilibrated field

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Error in the energy norm

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Error in the energy norm

Rewriting the problem in terms of the flux and the temperature

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Error in the energy norm

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Error in the energy norm

will fulfill exactly the first 2 equations. will fulfill exactly the 3rd_equation.

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Error in the energy norm

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Error in the energy norm

Formalizing this idea, it can be shown that

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Error in the energy norm

Expanding

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Error in the energy norm

Expanding

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Goal oriented error estimation

The error in energy norm is not always relevant.

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Goal oriented error estimation

Goal: Bound for the quantity of interest

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Goal oriented error estimation

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Goal oriented error estimation

Cauchy-Schwarz inequality

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Goal oriented error estimation

Cauchy-Schwarz inequality

Use the bound in the energy norm,

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More bounds

It is possible to lower bound the error in energy norm

Sharper bounds for the quantity of interest can be obtained through the use of polarisation identity

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Validation

The “exact” quantity of interest is computed with 512 MC realisations.

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Validation

The “exact” quantity of interest is computed with 512 MC realisations.

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Validation

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Validation

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Validation

Studied in a domain homogenised through rule of mixture. Dual problem

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Validation

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What if the bounds are not tight enough?

This is usually the case when the contrast is very high. Two possible solutions

● Adaptivity: solve in a certain subdomain the heterogeneous problem

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Enriched approximation

Idea: Solve RVEs, filter their solution

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Enriched approximation

Assembling the system of equations, 3 types of terms appear

Idea: We do not need to solve the RVE for all particle layouts, we

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Enriched approximation

Idea: We do not need to solve the RVE for all particle layouts, we

only need to compute

Remarks:

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Enriched approximation

Preliminary results

10% reduction

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Summary

– A method to estimate error in homogenisation was presented • Represent the heterogeneous problem through an SPDE

• A posteriori error estimation tools used to compute the error • The computation of the bound is deterministic

• The second moment of the quantity of interest can be bounded – On going work: Making the bounds sharper

• Through adaptivity

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References

– P Ladeveze, D Leguillon. Error estimate procedure in the finite

element method and applications. SIAM Journal on Numerical Analysis, 1983

– JT Oden and KS Vemaganti. Estimation of local modelling error and goal oriented adaptive modelling of heterogeneous materials. Journal of Computational Physics, 2000

– JP Moitinho de Almeida, JA Teixeira de Freitas. Alternative approach to the formulation of hybrid equilibrium finite elements. Computers & Structures, 1991