Summation rules for higher order Quasi-continuum methods

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Quasi-continuum methods

Claire Heaney, Lars Beex, St´ephane Bordas and Pierre Kerfriden

Institute of Mechanics and Advanced Materials

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1 Lattice models

Atomistic lattice models Structural lattice models

2 The Quasi-continuum method

Formulation Approximations Summation rules

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Atomistic lattice models

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separation of atoms

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Structural lattice models

Lattice models comprising springs or beams have been used to model atomistic crystalline materials

fibrous materials collagen networks heterogenous materials

Quasi-continuum method:

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Structural lattice models

Lattice models comprising springs or beams have been used to model atomistic crystalline materials

fibrous materials collagen networks heterogenous materials

Quasi-continuum method:

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Formulation

For displacements and rotations, aT_{= {u}

i, vi, wi, θix, θyi, θiz}

nN odes

i=1 ,

we minimise the potential energy of the system: arg min(Eint(a) − fext· a), where Eint=

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Formulation

For displacements and rotations, aT_{= {u}

i, vi, wi, θix, θyi, θiz}

nN odes

i=1 ,

we minimise the potential energy of the system: arg min(Eint(a) − fext· a), where Eint=

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Approximations in the coarse region

Efficiencies made in the coarse region:

1 “interpolation”: number of degrees of freedom are reduced by introducing representative lattice nodes or “rep-atoms”

a_rT_{= {u}

i, vi, wi, θxi, θiy, θzi}

rN odes

i=1 ,

2 “summation rule”: find representative interactions or sampling

beams

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Approximations in the coarse region

Efficiencies made in the coarse region:

1 “interpolation”: number of degrees of freedom are reduced by introducing representative lattice nodes or “rep-atoms”

a_rT_{= {u}

i, vi, wi, θxi, θiy, θzi}

rN odes

i=1 ,

2 “summation rule”: find representative interactions or sampling beams

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Advantages of QC

lattice defects can be modelled accurately (in the fully resolved region)

no continuum model is required in the coarse region, as the approximation is based on the lattice model itself

Disadvantage of QC

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Advantages of QC

lattice defects can be modelled accurately (in the fully resolved region)

no continuum model is required in the coarse region, as the approximation is based on the lattice model itself

Disadvantage of QC

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Summation rules

Summation rules - guided by Gaussian quadrature rules. Three rules:

closest summation rule (implemented in Beex et. al. 2014) mid-beam rules

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Summation rules

closest summation rule (implemented in Beex et. al. 2014)

mid-beam rules

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Summation rules

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Summation rules

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Uniaxial test [0,200] × [0,100] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 | energy - energy all | / energy all

closest summation rule

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Uniaxial test [0,200] × [0,100] 10-6 10-5 10-4 10-3 10-2 10-1 | energy - energy all | / energy all

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Uniaxial test [0,100] × [0,50] 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 | energy - energy all | / energy all

closest summation rule

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Uniaxial test [0,100] × [0,50] 10-5 10-4 10-3 10-2 10-1 100 | energy - energy all | / energy all

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Mesh quality, mesh 4 on [0,100] × [0,50] (closest beam rule) 7.327.74 8.158.56 8.979.38 9.7910.20 10.61 1.05 1.11 1.17 1.23 1.29 1.35 1.41 1.47 1.53 1.59 0 5 10 15 20 all elements 7.327.74 8.158.56 8.979.38 9.7910.20 10.61 1.05 1.11 1.17 1.23 1.29 1.35 1.41 1.47 1.53 1.59 0 5 10 15 20

those elements whose beams extend outside the element

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Mesh quality, mesh 5 on [0,100] × [0,50] (closest beam rule) 4.605.00 5.405.80 6.206.60 7.007.40 7.80 1.06 1.17 1.28 1.39 1.49 1.60 1.71 1.82 1.92 2.03 0 10 20 30 40 all elements 4.605.00 5.405.80 6.206.60 7.007.40 7.80 1.06 1.17 1.28 1.39 1.49 1.60 1.71 1.82 1.92 2.03 0 10 20 30 40

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Mesh quality, mesh 4 on [0,100] × [0,50] (mid-beam rule, local) 7.327.74 8.158.56 8.979.38 9.7910.20 10.61 1.05 1.11 1.17 1.23 1.29 1.35 1.41 1.47 1.53 1.59 0 5 10 15 20 all elements 7.327.74 8.158.56 8.979.38 9.7910.20 10.61 1.05 1.11 1.17 1.23 1.29 1.35 1.41 1.47 1.53 1.59 0 5 10 15 20

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Mesh quality, mesh 5 on [0,100] × [0,50] (mid-beam rule, local) 4.605.00 5.405.80 6.206.60 7.007.40 7.80 1.06 1.17 1.28 1.39 1.49 1.60 1.71 1.82 1.92 2.03 0 10 20 30 40 all elements 4.605.00 5.405.80 6.206.60 7.007.40 7.80 1.06 1.17 1.28 1.39 1.49 1.60 1.71 1.82 1.92 2.03 0 10 20 30 40

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The results show that

the mid-beam rules have a significantly lower error than the closest summation rule

of the two mid-beam rules, the local rule performs better than the non-local rule

Current work involves

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EPSRC EP/J01947X/1

Towards rationalised computational expense for simulating fracture over multiple scales.

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Euler-Bernoulli beams

The internal energy of beam i given in a local coordinate system

Ei = E 2 Z V ε_xx/u+ ε_xx/by+ ε_xx/bz2 + γ 2 xy+ γ 2 xz 2(1 + ν) dV = E 2 Z V ub− ua L − yv ′′_{− zw}′′ 2 + θ x b − θax L 2 y2 + z2 2(1 + ν) dV . We expand v(x) and w(x) as third degree polynomials and calculate their coefficients by using the following conditions

v(x = 0) = va w(x = 0) = wa

v(x = L) = vb w(x = L) = wb

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Meshes

Number of triangles out of which sampling beams stray, for the domain [0, 200] × [0, 100].

mesh # number of closest rule mid-beam rule mid-beam) rule

triangles (non-local) (local)

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Meshes

Number of sampling beams for the domain [0, 200] × [0, 100]. Total number of beams in the lattice: 40300.

mesh # number of closest rule mid-beam rules

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Meshes

Total degrees of freedom of the lattice for the domain [0, 200] × [0, 100]: 121806.

mesh # number of triangles degrees of freedom