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University of Liège
Aerospace & Mechanical Engineering
Alternative numerical method in continuum mechanics
COMPUTATIONAL MULTISCALE
Van Dung NGUYEN
Innocent NIYONZIMA
Aerospace & Mechanical engineering
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Content
• Introduction
• Basic equations
– Basic assumptions
– Definition of microscopic problem
– Coupling of microscopic and macroscopic problem
• Finite element implementation
• Numerical examples
– Mechanical example
– Magneto-static example
• Limitations
• Perspectives
2010-20 11 Content 2
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Scales of complex micro-structured materials
Batch foaming - PHTR46B4
2010-2011 Introduction 3
Carbon nanotubes
manufactured by nanocyl
Scales of complex micro-structured materials
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Classification of multiscale methods
• Multiscale problems can be divided into two classes :
– Type A problems : deal with isolated defects near which the macroscopic
models are invalid (shocks, cracks, dislocations,…). Elsewhere the explicitly
given macroscale model is valid.
– Type B problems : constitutive modeling is based on the microscopic
models for which the macroscopic model is not explicitly available and is
instead determined from the microscopic model.
• Heterogeneous Multiscale Method (HMM) has been attempting tobuild a unified framework for designing effective simulation methods that
2010-2011 Introduction 4
build a unified framework for designing effective simulation methods thatcouple microscale and macroscale models. HMM applies for both type Aand type B problems
• In general, there is no restriction on models that can be used at bothmacroscale and microscale (continuum media, molecular dynamics,quantum physics,…).
• In this presentation, we restrict ourselves to type B problems. We alsoconsider continuum media models at both scales.
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Why use computational multiscale methods?
• The heterogeneous nature and the multiscale structure of complexmicro-structured materials confer them some remarkable properties:
– mechanical: resistance, increased Young modulus, negative Poisson ratio…
– electromagnetic/optic: resistance, electromagnetic shielding, µ < 0, ε < 0…
• Classical methods are not efficient to model these materials :
– direct simulation methods such as the finite element method are costly in
Computational multiscale methods allow to model complex micro-structured
materials using more accurate macroscopic constitutive laws.
2010-2011 Introduction 5
– direct simulation methods such as the finite element method are costly in
terms of computational time and memory.
– the trial-and-error approach that consists in manufacturing the material and
then measure its physical properties is costly and not suited for optimization.
• Computational methods have been succesfully used to model complexmicro-structured materials :
– for linear/slightly nonlinear materials,the Mean-Field Homogenization
method is efficient.
– for highly nonlinear complex micro-structured materials, all other methods
fail and only Computational Multiscale Methods (CMM) remain valid.
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• In the CMM framework, 2 problems are defined:
• Macroscale problem
• Microscale problem (Boundary Value Problem- BVP)
• Scale transitions allow coupling two scales :
– upscaling: constitutive law (e.g.: stress, tangent operator) for macroscale
problem is determined from microscale problem (e.g. using averaging theory).
– downscaling: transfer of macroscale quantities (e.g.: strain) to the microscale.
These quantities allow determining equilibrium state of BVP.
CMM – macro/micro-problems and scale transitions
2010-2011 Introduction 6
Scale transitions
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CMM – Basic assumptions
• Material macroscopically sufficiently homogeneous, but microscopicallyheterogeneous (inclusions, grains, interfaces, cavities,…) � (continuummedia and averaging theorems).
• Scale separation: the characteristic at microscale must be muchsmaller than the characteristic size at macroscale.
• Two additional assumptions can be made:
– The characteristic size of the heterogeneities must be much greater than the
molecular dimension (continuum media at the microscale).
– The characteristic size of the heterogeneities must be much smaller than the– The characteristic size of the heterogeneities must be much smaller than the
size of the RVE (RVE statistically representative).
2010-2011 Introduction 7
Heterogeneous micro-structure associated
with macroscopic continuum point M
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Some key advantages:
– Do not require any constitutive assumption with respect to the overall
material behavior.
– Enable the incorporation of large deformations and rotations on both micro
and macro-level.
– Are suitable for arbitrary material behavior, including physically non-linear
and time dependent behavior.
– Provide the possibility to introduce detailed micro-structural information,
including a physical and/or geometrical evolution of the microstructure, in the
macroscopic analysis.
Computational Multiscale Method – advantages
macroscopic analysis.
– Allow the use of any modeling technique at the micro-level.
– Microscale problems are solved independently from each others and the
method can be easily parallelized.
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• Approaches to solve scale transitions problems:
– 1965: Method based on Eshelby results: Mean-field homogenization (Hill)
– 1978: Asymptotic homogenization method (A. Bensoussan et al.)
– 1985: Global-local method (Suquet).
– 1995: First-order computational homogenization method (Ghosh et al.)
– 2001: Extension to the second-order (Geers et al.)
– 2003: Heterogeneous Multiscale Method - HMM (E et al.).
– 2007: Continuous–discontinuous multiscale approach (Massart et al.) and
computational homogenization of thin sheets and shells (Geers et al)
History
computational homogenization of thin sheets and shells (Geers et al)
– 2008: Thermo-mechanical coupling (Ozdemir et al.) and Computational
homogenization of interface problems (Matous et al)
• Applications:
– Mechanics: damage and fracture analysis, thin sheet and shells, failure
analysis of cohesive interfaces, flow transport through heterogeneous porous
media…
– Heat transfer: heat conduction in heterogeneous media
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Macroscopic problem
• Macroscopic problem
– Weak form
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Definition of microscopic problem
• Representative volume element (RVE)
– A model of material micro-structure which is used to obtain the macroscopic
material response at a macroscopic material point.
• Selection of RVE
– RVE contains all necessary information of micro-structure
RVE associated with
a macroscopic point
– RVE contains all necessary information of micro-structure
– Computation efficiency
• RVE equilibrium state
– In absence of body forces:
– Constitutive law:
– For hyper-elastic material
– Equilibrium state of the RVE is assumed to be consistent with the boundary
condition, which are related to the macroscopic strain field
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Definition of microscopic problem
• Boundary condition
– Strain driven problem
– Microscopic strain:
– Macroscopic strain:
– Using Gauss theorem:
– Split of displacement field: mean part and fluctuation part:
– Constrain on the fluctuation field:
– Boundary condition must be defined to satisfy (*)
2010-20 11 Basic equations 12
(*)
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Definition of microscopic problem
• Boundary condition
– Hill assumption (rule of mixtures): no fluctuations in RVE
– Linear displacement boundary condition (Dirichlet boundary condition): no
fluctuation at RVE boundary
– Periodic boundary condition: periodicity of fluctuation field and anti-
periodicity of traction field at RVE boundary
– Minimal kinematic boundary condition (Neumann boundary condition)
2010-20 11 Basic equations 13
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Definition of microscopic problem
• Boundary condition
– Periodic boundary condition is the most efficient in terms of convergence
rate
– Linear displacement upper-estimate the effective properties
– Constant traction (Neumann BC) under-estimate the effective properties
2010-20 11 Basic equations 14
RVE selection Convergence of average properties
with increasing RVE size.
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Coupling of microscopic and macroscopic problem
• Strain averaging
• Hill-Mandel principle:
– Energy consistency in the transition of macro- and micro-scales:
(**)
– For elastic material in small strain:
– Virtual strain:
– Equation (**) becomes:
– Stress averaging:
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Coupling of microscopic and macroscopic problem
• Hill-Mandel principle:
– Hill-Mandel condition in terms of fluctuation part:
– Using the equilibrium state and Gauss theorem:
– All boundary conditions previously defined satisfy the condition (***)
(***)
• Stress averaging and tangent operator
– Equilibrium state and Gauss theorem:
– Tangent operator:
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Finite element implementation
• Finite element model at microscopic scale
– Minimal potential energy principle
– Discretize the displacement fluctuations at element level
– Assemble operator:– Assemble operator:
– Approximation of internal energy:
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Finite element implementation
• RVE boundary condition
– Linear displacement boundary condition
• For M nodes on RVE boundary
• Partitioning of fluctuation field on RVE boundary
• Linear constraints on fluctuation displacement
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Finite element implementation
• RVE boundary condition
– Constant traction boundary condition
• From strain averaging equation
• Assemble on RVE boundary elements
(*)
• Linear constraints on fluctuation displacement
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Finite element implementation
• RVE boundary condition
– Periodic boundary condition
• Periodic mesh requirement
• Periodic mesh: apply on matching node on RVE boundary
Periodic mesh and non-periodic mesh
• Linear constraints on fluctuation displacement
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Matching node
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Finite element implementation
• RVE boundary condition
– Periodic boundary condition
• Non-periodic mesh: polynomial interpolation method
• Linear constraints on fluctuation displacement
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Finite element implementation
• RVE equilibrium state
– Minimize
– Subject to:
• Enforcement by Lagrange multipliers
– Lagrange function
– Equilibrium state
– Internal force
– Nonlinear system
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Finite element implementation
• Enforcement by Lagrange multipliers
– Nonlinear system to solve
– Solve by Newton-Raphson procedure
• Step 0
• Step 1
• Step 2
• Step 3
• If EXIT
• else GOTO step 1
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Finite element implementation
• Enforcement by constraint elimination
– Problem:
– Decomposition: dependent part and independent part from constraints
– New equation of independent part– New equation of independent part
– Solve by Newton-Raphon procedure
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Nested solution scheme
MACRO
– Step 1: Initialization
• Assign RVE to every
integration points (IPs)
• Set for all Ips
– Step 2: next load increment
at macroscopic problem
MICRO
– RVE analysis
• Prescribe boundary condition
• Calculate homogenized
tangent operator
– Step 3: next Iteration
• Solve macroscopic problem
• Calculate macroscopic
forces
– Step 4: convergence
• If convergence � step 2
• Else � step 3
– RVE analysis
• Prescribe boundary condition
• Calculate homogenized
stress
• Calculate homogenized
tangent operator
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Numerical examples
• ProblemVertical displacement
2010-20 11 Mechanical example 26
• Material
– Matrix:
– Inclusion:
Finite element meshPlane strain problem
and boundary condition
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Numerical examples
• RVE analysis
– Macroscopic strain
– Homogenized stress
• Linear displacement boundary condition
RVE finite element mesh
Inclusion
Matrix
2010-20 11 Mechanical example 27
• Periodic boundary condition
RVE finite element mesh
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Numerical examples
• RVE analysis
– Displacement field
2010-20 11 Mechanical example 28
– Von-Mises stress
Linear displacement BC Periodic BC
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Numerical examples
• Multi-scale analysisQuadratic triangle
2010-20 11 Mechanical example 29
Macroscopic finite element mesh
RVE finite element mesh
Inclusion
Matrix
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Numerical examples
• Multi-scale analysis
– Vertical displacement
Vertical displacement
2010-20 11 Mechanical example 30
– Reaction force
Plane strain problem
and boundary conditionReaction force
Linear displacement BC 3421.955
Periodic BC 3406.800
Single scale 3381.519
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Numerical examples
• Multi-scale analysis
– Von-Mises stress
2010-20 11 Mechanical example 31
Single scale Linear displacement BC Periodic BC
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Limitations
• The “scale separation” assumption sets limitations inherent to thecomputational multiscale method. Problems for which this assumption isinvalid can not be accurately modeled using these methods.
• The computational time is still long because a microscale problem mustbe solved in each gauss point. Micro-problems are solved independentlyand parallelization can greatly reduced the computational time.
• Limitations inherent to the first-order scheme have been addressed:
– Second-order schemes have been proposed to account for higher-order
deformation gradients at the macroscale and the size effects of the RVE. deformation gradients at the macroscale and the size effects of the RVE.
– The continuous-discontinuous approach has been proposed to deal with
problems of intense localization (e.g.: damage and fracture analysis).
– Computational homogenization of structured thin sheets and shells, based
on the application of second-order homogenization have been proposed.
– …
• Thermo-mechanical coupling has been addressed. Coupling with other physics (heat, mechanics, electromagnetism,…) is essential to fully characterize complexe behavior of multiscale materials.
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Perspectives
• Computational homogenization of emerging and evolving localizationbands on the macro-scale
• Multi-physics and coupled field problems (electro-mechanical, thermo-electrical, fluid-structure interaction, magneto-electro-elasticity,acoustics, etc.)
• Dynamic problems, including inertia effects and/or propagating waves
2010-20 11 Perspectives 33
• Problems related to non-convexity and microstructure evolutionemanating from the micro-scale
• Integration of phase field models across the scales.
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References
• Computational homogenization of emerging and evolving localizationbands on the macro-scale
• Multi-physics and coupled field problems (electro-mechanical, thermo-electrical, fluid-structure interaction, magneto-electro-elasticity,acoustics, etc.)
• Dynamic problems, including inertia effects and/or propagating waves
2010-20 11 Perspectives 34
• Problems related to non-convexity and microstructure evolutionemanating from the micro-scale
• Integration of phase field models across the scales.
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References
2010-2011 Vehicle Safety: Numerical simulations 35