Reduced order modelling: towards tractable computational homogenisation schemes

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& Advanced Materials

Reduced order modelling: towards tractable

computa5onal homogenisa5on schemes

Pierre Kerfriden1,_*,

Olivier Goury1_{,
Chi
Hoang}1_,

Juan José Ródenas2_{,
Timon
Rabczuk}3_{,
Stéphane
Bordas}4,1

1_{Cardiﬀ
University,
UK}

2_{Bauhaus-‐Universität
Weimar,
Germany} 3_{Universidad
Politecnica
de
Valencia,
Spain}

(2)

Ins$tute of Mechanics & Advanced Materials

• Eﬃcient numerical predic$on of material and structural failure

• Characterisa$on and op$misa$on of composites

Team in Cardiﬀ: main research topics

[Silani et al., 2013] Initial crack

distribution Final fracture _{[Sutula
et
al.,
2013]}

[Kerfriden et al., 2010]

S.P.-‐A. Bordas

(3)

• Interac$ve simula$ons of biological structures

• Simpliﬁed Link between CAD/CT scans and analysis

Team in Cardiﬀ: main research topics

A 4.30 A 10 7 2.6_⇥ U Q 0 0.02 0.04 0.06 0.08 0.1 0 20 40 60 80 100 120 140 U Q 15% 3.4 U Q xc Wc 1% 25 U

[Courtecuisse et al., 2013]

[Sco` et al., 2013]

show the flexibility of the non-conforming coupling, the solid part was moved slightly to the right and the deformed configuration is given in Fig. 34. The same discretisation for the plate is used. This should serve as a prototype for model adaptivity analyses to be presented in a forthcoming contribution.

Figure 33: Square plate enriched by a solid: transverse displacement plot on deformed configurations of plate model (left) and solid-plate model (right).

Figure 34: Square plate enriched by a solid: transverse displacement plot where the solid part was moved slightly to the right .

7. Conclusions

We presented a Nitsche’s method to couple (1) two dimensional continua and beams and (2) three dimensional continua and plates. A detailed implementation of those coupling methods was given. Numerical examples using low order Lagrange finite elements and high order B-spline/NURBS isogeometric finite elements provided demonstrate the good performance of the method and its versatility. Both classical beam/plate theories and first order shear beam/plate models were addressed. Conforming coupling where the continuum mesh and the beam/plate mesh is not overlapped

37

[Nguyen et al., 2013]

Local 3D analysis

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• Advanced discre$za$on techniques for complex PDEs

§  XFEM/meshfree

§  Isogemetric analysis

Toolkit 1: advanced discre$sa$on methods

[Bordas et al., 2008] Taylor bar problem

(dynamic fragmenta$on)

b)

Figure 5 Two main trivariate NURBS patches with parametric directions: a) brick to represent beam; b) cylinder to represent added mass

Modelling the six patches with their own dimensions to represent the beams and the added masses, rigid roto-translation of the control points allow the patches to be oriented and repositioned in order to compose the desired configuration of the assembly (Figure 6). The same roto-translation process will be used to orient the second beam to represent the different parametric configurations.

Figure 6 IGA model

In Figure 7 is intended to underline the geometrical comparison among the CAD model, the LUPOS model and the IGA model, to check for geometric consistency.

the minimum number for representing with enough accuracy the bending of higher modes. The functions are cubic in all the three parametric dimensions.

For the added masses, a single element would be enough, considering that the they can be considered infinitely rigid with negligible contribution to the elasticity. Due to Nitsche coupling, a finer discretization is necessary for better let the method distribute the coupling entries on more elements. The patches have cubic functions for all the parametric dimensions.

Figure 9 Refined model used to perform the simulation

In Figure 10 it is shown the comparison of Mode 3 and Mode 6 using LUPOS, a tri-dimensional standard FEM and IGA, for the 30° configuration.

single element through the thickness and one element along the width are enough, while for the length five elements are the minimum number for representing with enough accuracy the bending of higher modes. The functions are cubic in all the three parametric dimensions.

For the added masses, a single element would be enough, considering that the they can be considered infinitely rigid with negligible contribution to the elasticity. Due to Nitsche coupling, a finer discretization is necessary for better let the method distribute the coupling entries on more elements. The patches have cubic functions for all the parametric dimensions.

Figure 9 Refined model used to perform the simulation

In Figure 10 it is shown the comparison of Mode 3 and Mode 6 using LUPOS, a tri-dimensional standard FEM and IGA, for the 30° configuration.

Model simpliﬁca$on (CAD) IGA

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[Akbari et al., 2013]

• Mul$level methods to reduce CPU $me by orders of magnitude and devise robust, eﬃcient code/model coupling

§  HPC Adap$ve mul$scale models/solvers

with controlled accuracy

Toolkit 2: mul$level methods

cohesive interfaces (damage model)

orthotropic linear plies

laminate (carbon/ epoxy) bolt (steel)

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• Mul$level methods to reduce CPU $me by orders of magnitude and devise robust, eﬃcient code/model coupling

§  Virtual chart with controlled

accuracy via ROM for mul$scale

modelling and real-‐$me op$misa$on

Toolkit 2: mul$level methods

[Kerfriden et al., 2013]

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y z Sampling “Online” (evaluation) for a particular parameter Optimal linear combination “Offline” (training) for all parameters

SVD

Direct evaluation

KA KA0

“oﬄine” / “online” strategy

above calculations are thus multiplied by

M times. In summary, for the offline stage, the

operation counts depend on N and hence, its computational cost is very expensive.

In the online stage – performed many times, for each new parameter µ – we first assemble the

RB primal matrices in (53), this requires O!Npr2(Qm+ Qc+ Qa)

"

operations. We then solve the RB

primal equation (51), the operation counts are O(Npr3 + KNpr2 ) as the RB matrices are generally full.

Finally, we evaluate the output displacement sN(µ, tk) from (52) at the cost of O(KNpr). Therefore,

as required in real-time context, the online complexity is independent of _{N , and since N}pr ! N we

can expect significant computational saving in the online stage relative to the classical FE approach.

5 Numerical example

In this section, we will verify both POD–Greedy algorithms by investigating an numerical example which is a three-dimensional dental implant model problem in the time domain. This model problem is similar to that in the work of Hoang et al. [5]. The details are described in the following.

5.1 A 3D dental implant model problem

(a) Z Y X X Y Z (b)

Figure 1: (a) The 3d simplified FEM model with sectional view, and (b) meshing in ABAQUS. We consider a simplified 3D dental implant-bone model in Fig.1(a). The geometry of the simplified dental implant-bone model is constructed by using SolidWorks 2010. The physical domain Ω consists

of five regions: the outermost cortical bone Ω1, the cancellous bone Ω2, the interfacial tissue Ω3, the

dental implant Ω4 and the stainless steel screw Ω5. The 3D simplified model is then meshed and

analyzed in the software ABAQUS/CAE version 6.10-1 (Fig.1(b)). A dynamic force opposite to the

x−direction is then applied to a prescribed area on the body of the screw as shown in Fig.2(a). As

mentioned in Section 3.3, all computations and simulations will be performed for the unit input loading case, since other input loading cases can be easily inferred from the Duhamel’s convolution. We show on Fig.2(b) the time history of an arbitrary loading case which we will show its Duhamel’s convolution later. The output of interest is defined as the average displacement responses of a prescribed area on

the head of the screw (Fig.2(a)). The Dirichlet boundary condition (∂ΩD) is specified in the

bottom-half of the simplified model as illustrated in Fig.2(a). The finite element mesh consists of 9479 nodes and 50388 four-node tetrahedral solid elements. The coinciding nodes of the contact surfaces between

16

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•  Introduc$on: computa$onal homogenisa$on •  Virtual charts for parametrised homogenisa$on

in linear elas$city

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• Bo`om-‐up view: replace heterogeneous

subscale model by an equivalent, smoother, model at the scale where predic$ons are

required (i.e. macroscopic scale)

• When is scale-‐bridging necessary?

§  Derive predic$ve macroscopic models that are diﬃcult to obtain using phenomenological approaches

§  Op$mise subscale proper$es to obtain be`er overall characteris$cs

§  Observa$ons at microscale but approxima$ons required away from region of interest to

remain tractable

Mul$scale modelling

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Examples

18

Fig.12 Optimal distribution of reinforcing ingredients considering area of interest #2 (a), objective

function versus iterations (b)

Fig.13 Shear stress profile for area #2 considering homogeneous and optimal distribution of

reinforcements

In the next case, area of interest #3 is defined on the core of the beam and includes central elements of the core as illustrated in Fig.9. The same types of results are presented in Fig.14 (a) and 14(b).

Fig.14 Optimal distribution of reinforcing ingredients considering area of interest #3 (a), objective

function versus iterations (b)

Op$misa$on of ﬁber content in sandwich beams to minimise interfacial stress in

Approxima$on of the

behaviour of polycrystalline materials away from

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• Knowing the governing equa$ons at the microscale, can we ﬁnd homogeneous

governing equa$ons at the macroscale s. t.:

§  The solu$on of the macroscale problem converges to the solu$on of the microscale problem when the scale ra$o tends to zero

• Hopefully: the solu$on of the macroscale

problem is a good approxima$on of the solu$on of the microscale problem (in some sense)

even when the scale ra5o is not very small.

Mathema$cal formula$on

Homogenisa$on (Or discrete model) Heterog. con$nuum PDE with constant coeﬀ.

Error in QoI macroscopic < Tolerance

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• Heterogeneous microstructure undergoing moderate deforma$ons, observa$ons at macroscale, slow loading, scale separability • Macroscale candidate model: lin. elas$city

§  Equilibrium

§  Kinema$c equa$ons

§  Cons$tu$ve rela$on by classical micromechanics

Classical homogenisa$on in linearised elas$city

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• A`ach a representa$ve volume element to the material point: volume of material large enough to represent the sta$s$cs of the distribu$on

of material proper$es (unit cell in periodic case)

• Suppose that the RVE is mechanically equilibrated:

• The eﬀec$ve cons$tu$ve law the/a rela$onship between average stress and average strain

Classical homogenisa$on in linearised elas$city

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• Obtain by solving RVE problem numerically

§  Ill-‐posed, requires BC for ﬂuctua$on compa$ble with

§  One possibility: Dirichlet problem, ﬂuctua$on vanishes on boundary →  Very expensive too solve

Computa$onal homogenisa$on

Elas$city, constant

coeﬃcients +

Homogeneous strain

(macroscopic part) “micro” ﬂuctua$on

< m >= _SM < ✏m >

< ✏(_{eu(y) >} ¯

(14)

•  Introduc$on

•  Virtual charts for parametrised homogenisa$on in linear elas$city

(15)

• Homogenisa$on

• Dirichlet localisa$on problem

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• Homogenised Hooke tensor (assuming major and minor symmetries)

(17)

• Response surface method: interpolate the quan$$es of interest

Virtual charts

0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 P1 response surface Elas$c contrast Elas$c contrast

Elas$c contrast (log)

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• Interpolate the state variables instead: reduced order model

§  For any applied macroscopic ﬁeld, and any elas5c contrast

ROM-‐based Virtual charts

1⇤ + 2⇤ + 3⇤ “Macro ﬁeld” “Micro ﬂuctua$on” u(µ) _⇡ +↵1(µ)⇥ +↵2(µ)⇥ +... 1 2

Generalised coordinate: minimise an error measure (Galerkin)

Kr(µ) ↵(µ) = Fr(µ)

K(µ) U(µ) = F(µ)

Kr_ij(µ) = T_i K(µ) _j

Reduced s$ﬀness

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• Interpolate the state variables instead: reduced order model

§  For any applied macroscopic ﬁeld, and any elas5c contrast

→  Osen, state variables belong to a low-‐dimensional vector space §  Then, post-‐process the quan55es of interest

→  (Sub-‐) op$mal surrogates can be constructed →  Error es$mates available

ROM-‐based Virtual charts

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• Separa$on of variables:

• Example of op$mal decomposi$on: POD

§  Not computable without approxima$on →  Galerkin Empirical POD

→  Reduced Basis method →  PGD

Reduced order model

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• Subop$mal surrogate in two steps:

§  Empirical POD “oﬄine”:

-  Compute sampling (Snapshot): -  Spectral analysis (SVD)

§  “Online” op5mal coordinates by Galerkin

Galerkin-‐POD with QRandom sampling

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y z Sampling “Online” (evaluation) for a particular parameter Optimal linear combination “Offline” (training) for all parameters

(22)

• Key issue 1: Empirical POD

§  Hierarchical sampling too expensive (curse of dimensionality) →  Quasi-‐random sampling

→  Error evaluated by cross-‐valida$on es$mates and stagna$on criteria

Key issue 1: choice of samples

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Convergence es$mate

0 2 4 6 8 10 −6 −5 −4 −3 −2 −1 0 1 0 2 4 6 8 10 −6 −5 −4 −3 −2 −1 0 1 Max Re la$ ve e rr or in Max Re la$ ve e rr or in

Order of POD surrogate

˜ Q 1 ˜ Q 1

LOOCV LOOCV reduced training set

LOOCV reduced valida$on set

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• Greedy sampling (RBM) (see the work of Maday, Patera, …)

§  Evaluate the error of a Galerkin-‐surrogate of order N on a ﬁne grid of the parameter domain

§  Add to the basis the sample for which the predic$on error (in QoI) is the largest

→  Eﬃcient when sharp es$mate available →  But limited to small number of

parameters

• QRandom-‐based Greedy-‐sampling (see [Rozza et al. 2011]) • Op$misa$on-‐based Greedy sampling

(work of Volkwein, Willcox, …)

§  Look for a new sample to add to a basis N such that the error of the surrogate of order N+1 is minimised

→  Localisa$on of local minima is diﬃcult

Alterna$ve methods for snapshot selec$on

µ1

µ2

µ1

(25)

Kr(µ) ↵(µ) = Fr(µ)

• “online” cost:

But assembly cost needs to be small!

• OK if data is separable:

→ 

→  Otherwise, one needs further approxima5ons (force data separa$on)

Key issue 2: eﬃcient assembly of ROM

(26)

• Aim: Bound from above and below • Adjoint problem for each QoI:

→  Galerkin-‐POD:

§  is minus the residual of the forward problem in (computable) §  is the error in the ith_{adjoint
problem}

• Special case: compliant problem (adjoint and forward solu$ons i are collinear)

→ 

Goal-‐oriented error es$ma$on

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• Upper bound in energy norm: Error in the cons$tu$ve rela$on [Ladevèze ’85][Ladevèze and Chamoin ‘11]

§  The recovered stress must be sta$cally admissible in the FE sense

→  More eﬀec$ve as the recovered stress approaches the FE stress

Error bounding by duality

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• Construc$on of the sta$s$cally admissible stress: POD-‐surrogate: SVD from stress snapshot:

§  “Oﬄine:” Empirical POD on stress samples

→  Basis such that average projec$on error over snapshots is minimised.

§  “Online:” Minimisa$on of CRE upper bound:

Local error es$ma$on

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Schema$c of duality-‐based ROM

proposed to estimate this quantity whilst retaining the bounding property. However, coercivity lower bounds can be pessimistic in the case of elasticity. In this paper, we propose to proceed di↵erently by using the concept of the Constitutive Relation Error (CRE) [32], which only requires to manipulate the concepts of displacement and stress admissibilities. In particular, no coercivity bound is required. The CRE is a widely used technique to bound the error associated to a displacement-based approxi-mation of elasticity problems. In particular, it has been applied to the evaluation of discretisation errors in a finite element context [33], where it coincides in practical implementations with the Equilibrated Residual approach (see for instance [52, 3, 50, 14]). Conceptually, the CRE proposes to construct a recovered stress field that is statically admissible, or equilibrated. Applying the constitutive relation to the kinematically admissible finite element solution that needs to be verified, one obtains a non-equilibrated stress field, called finite element stress field. The distance (in energy norm) between the recovered stress field and the finite element stress field is a bound for the discretisation error. All the technical difficulty resides in the construction of the equilibrated stress field [45].

... ... ... −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x y z Sampling (Snapshot) Constitutive relation U p p er b ou n d Er ror FE Solution to approximate ``online” Minimisation Potential energy Minimisation complementary energy ⇣ uh_(µs ns), h_(µs ns) ⌘ ⇣ uh_(µs 1), h_(µs 1) ⌘ ⇣ uh(µs₂), h(µs₂)⌘ 3 1 2 e nt+nb e 1 e2 e 1 e2 en˜ n 1 2 uh(µ), h(µ) r_(µ) µ ur(µ) = n X i=1 i ↵i(µ) + nw X i=1 i w i(µ) ... FE problem parametrised by Expl icit e_va lua tion

Explicit eva

luation b(µ) = ˜ n X i=1 e i↵ei(µ) +D e(µ0) : ✏ nt X i=1 e i t i(µ) + nb X i=1 e i+nt b i(µ) Re duc ed s pa ce for t he di spl ac em ent SVD SVD FE pre-computations Re duc ed s pa ce for t he s tre ss

Figure 1: Schematic representation of the Galerkin-POD error bounding method based on the Consti-tutive Relation Error.

We extend this idea to the certification of the Galerkin-POD, which will provide a bounding tech-nique that is conceptually simple to understand, implement and control. Here, the reference is the finite element solution. Therefore, we first redefine the notion of statical admissibility, and require the recovered stress field to verify the equilibrium in the finite element sense. Then, at any point of the parameter domain, we can upper bound the reduced order modelling error by measuring a

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• Illustra$on for and (compliant problem)

→  Lower bound necessary to control the sharpness of the upper bound as no a priori convergence es$mate available

Results

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 0.8 1 1.2 1.4 1.6 1.8 2 100 101 102

Surrogate order ra$os

Eﬀ ec $v ity inde x (log) Re la$ ve e rr or in (lo g) ˜ Q 1

Elas$c contrast (log)

✏M = ✓ 1 0 0 0 ◆ ˜ Q₁ _{⌘ D} e ? 1111 Increasing order of stress surrogate

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• (Sub-‐) op$mal virtual charts for parametrised homogenisa$on problems, based on Galerkin-‐POD

• No a priori assump5on on ﬂuctua5on ﬁelds: smooth transi$on

between approximate RVE problem and fully resolved (FEM), by just enriching the reduced basis

• Guaranteed accuracy

• We are working on op$mal sampling of displacement versus stress surrogates to reach the desired level of accuracy on QoI

→  Goal-‐oriented, hybrid reduced basis strategy

(32)

Reduced basis sampling technique

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −10 −9 −8 −7 −6 −5 −4 −3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −10 −9 −8 −7 −6 −5 −4 −3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −10 −9 −8 −7 −6 −5 −4 −3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −10 −9 −8 −7 −6 −5 −4 −3 Re la$ ve e rr or in e ne rg y no rm (l og )

Elas$c contrast (log) _{Elas$c
contrast
(log)} Max error

Max error

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• Interpreta$on of error bound as uncertainty on QoI • Illustra$on for a compliant problem (if ):

→  uncertainty reduced by error reduc$on (increase displacement

accuracy) or increase in bound eﬀec$vity (increase accuracy of stress) →  Hybrid reduced basis

GO hybrid reduced basis approach

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• Start with one random sample at and ini$alise the surrogates

§  KA Displacement: §  Equilibrated stress:

• Max uncertainty over parameter domain detected at

§  Check which of the enrichments create maximum reduc$on in uncertainty

§  Proceed un$l max uncertainty is small enough

GO hybrid reduced basis approach

8 µ 2 P, b(µ) = e₁ ↵e1(µ) 8µ 2 P, ur(µ) = ₁ ↵1(µ) + uh,p(µ) µ0 ﬂuctua$on at µ0 stress at µ0 µ1 ur(µ) = 2 X i=1 i ↵i(µ) + u h,p_(µ) or

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Illustra$on for compliant problem

Worst uncertainty, best reduced by enriching stress surrogate

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• Guaranteed accuracy limited to ellip$c and parabolic (e.g. viscoelas$city) problems, with a priori separa$on of variables

(problems with parametrised geometry or nonlineari$es rarely respect this condi$on)

• Open ques$ons / remarks

§  Error bounds can be obtained for “smooth” problems, in which case empirical, more general es$mates seem to give reliable results.

§  How to choose the QoI in imbricated problems (chains of approxima$ons)?

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•  Introduc$on

•  Virtual charts for parametrised homogenisa$on in linear elas$city

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• Implicitly deﬁned nonlinear homogenised cons$tu$ve rela$ons • Macro problem:

§  Equilibrium:

§  CR: (not speciﬁed explicitly)

• Micro problem: §  Equilibrium: §  CR: (known, non-‐linear) • Scale linking §  DBC §  Stress average:

Nonlinear comput. homogenisa$on by FE

2

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• Damageable elas$c heterogeneous

• Modelled by a network of Euler-‐Bernouilli beams with CR:

• Proper$es of resul$ng problem:

§  Damage depends on deforma$on -‐> Nonlinear

§  Irreversibility of damage -‐> pseudo-‐$me dependent

Damageable RVE

Severe damage 0 2 4 6 8 10 12 14 16 18 20 −6 −4 −2 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 18 20 −6 −4 −2 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 18 20 −6 −4 −2 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 18 20 −6 −4 −2 0 2 4 6 8 10

Realisation 1

Realisation 2

Particle

distribution

Projection of

the material

properties

Figure 5: Two realisations of the random distribution of the material properties. The inclusions

are spherical, and their projection onto the lattice structure is represented in black. The grey bars

correspond to the matrix of the particulate composite, while the light grey ones define the weak

interface between aggregates and matrix.

In order to describe the randomness of the model formally, we introduce the probability space

H = (⇥, F, P ) for the random distribution of the material properties of the lattice network. ⇥ is the

ensemble of outcomes of the random generation,

_{F is the corresponding Sigma-algebra of subsets of ⇥}

and P is the associated probability measure. Any random distribution is characterised by distribution

function (26), a fixed maximum inclusion diameter D

_Max

, a fixed phase ratio, fixed material properties

for each of the phases, and a given lattice network.

Two realisations of the random distribution of material properties are illustrated in figure 5.

2.4 Discretisation

We now present briefly the discretisation technique of the lattice balance equations (15). The

displace-ment of a point of the neutral fibre of beam (b) is approximated using a unique finite eledisplace-ment, of first

order degree for the normal displacement, and third order degree for the deflection:

✓

v

(b)

w

(b)

◆ (⇠) =

(b) v

(⇠)

(b) w

(⇠)

!

0 B

B

@

v

(b)

(0)

w

(b)

(0)

w

(b)

◆ (⇠) =

(b)

(⇠) V

(b)

,

where the matrix of shape functions is given by:

(⇠) =

0 B

B

@

v,1

(⇠)

0

_w,1

(⇠)

0

_w,2

(⇠)

v,2

(⇠)

0

_w,3

(⇠)

0

_w,4

(⇠)

1 C

C

A

T

with

8 >

>

<

>

:

v,1

(⇠) = 1

1 L

⇠

v,2

(⇠) = ⇠

w,1

(⇠) =

2 L

3

⇠

3

3 L

1 L

⇠

2

.

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11

where t is the depth of the continuous structure, which is not necessarily equal to the depth of the

beams t(b).

Unfortunately, this theory, as far as the authors know, has not been extended to damage mechanics. It su↵ers the limitation of classical homogenisation to non-softening behaviours. In the literature, the elastic constants are used as a starting point for a phenomenological damage model, whose parameters then need to be fitted to experimental results. We follow the same approach. The damage model used in our simulations is described in the next subsection.

2.2.3 Elastic-damage law

The model presented here is based on classical damage mechanics [37], applied to lattice structures. We introduce two di↵erent damage mechanisms, one acting in traction and the other one acting in bending. This assumption is consistent with the model used in [5], where it is argued that damage in traction corresponds to damage due to hydrostatic deformations, while damage due to bending corresponds to shear damage.

We postulate the following Helmholtz free energy per unit length of beam (b):

(¯✏, d) = 1 2 ✓ E(b)S(b)(1 dn) < v,⇠ >+ v,⇠ v,⇠2 +E(b)I(b)(1 dt)✓,⇠2 ⌘ . (21)

dn and dt are two damage variables ranging from 0 to 1. They account for the non-reversible softening

of the beam with increasing load, in respectively tension and bending. Compression in this model does not dissipate energy, which is mathematically introduced by making use of the positive part extractor

< . >+. We introduce the compact notation d = (dn dt)T.

The relationship between generalised stress and strain in beam (b) is obtained as follows: ✓ N M ◆ = @ @¯✏ = 0 @E(b)S(b)(1 dn) < v,⇠ >+ v,⇠ 0 0 E(b)I(b)(1 dt) 1 A✓v,⇠ ✓,⇠ ◆ . (22)

This state law is the nonlinear counterpart of the linear state law (19).

The second state law, which links the damage variables to dual driving thermodynamic forces, reads: ✓ Yn Yt ◆ = @ @d = 1 2 0 @E (b)_S(b)< v,⇠ >+ v,⇠ v,⇠2 E(b)I(b)✓,⇠2 1 A . (23)

At last, an evolution law is defined to fully define the damage evolution: Y (t) = max ⌧t ((Yn(⌧ )) ↵ + ( Yt(⌧ ))↵) 1 ↵ _, ₍₂₄₎ dn = dt = min ✓✓ n n + 1 < Y Y0 >+ Yc Y0 ◆n , 1 ◆ . (25)

This damage law is inspired by the model described in [1] for composite laminates. We refer to this work for a comprehensive interpretation of the di↵erent parameters. We will just notice that Y is an equivalent damage energy release rate which governs the evolution of damage with traction and

banding. The critical value Yc is therefore the “strength” of the beam section.

2.3 Randomly distributed material properties

The damageable lattice model is used to derive a three-phase model for concrete. Such models consider three di↵erent entities: matrix (cement), inclusions (hard particles, assumed spherical) and an interface between these two entities (see [6] for an evidence of the existence of such interface). Plane (O, x, y) is a section of the three dimensional particulate composite structure. A projection of the material properties onto the lattice model is performed as follows:

9

“d” Damage variable represents the extent of subscale networks of cracks

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• Homogenisa$on using Dirichlet localisa$on on RVE:

§  Fully discrete equilibrium, for any load history:

§  Uniform displacement boundary condi$ons

Parametrised RVE problem

Parameters

(“ Dimension 3N “₎

U(t) = ¯U(t) + eU(t) N (x) ¯U(t) =

✓ ✏M₁₁(t) ✏M₁₂(t) ✏M₁₂(t) ✏M₂₂(t) ◆ · (x xc) Such that Fluctua$on vanishes on boundary = +

Full micro solu$on, used to

extract average stress

8 t 2 {t1, t2, ... , tN}, U? T F_int

⇣ e

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• Approxima$on for the ﬂuctua$on ﬁeld for any load history

• Galerkin:

• Solu$on by Newton:

Galerkin-‐ROM

Modal coordinates associated with POD basis vectors

8i

Requires integra5on, complexity depends

on the underlying discre$sa$on

U(t) _{⇡ U}r(t) :=

n

X

i=1

i ↵i(t) + ¯U(t)

Global basis func$ons for ﬂuctua$on ﬁeld (obtained by Snapshot POD)

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Only the contribu$ons of sample points are required

• Aﬃne approximate of the nonlinear force vector

• Coeﬃcients of the expansion obtained op$mally w.r. to sample points:

Gappy approxima$on of nonlinear term

Any varia$on around the Galerkin solu$on F_int (↵(t) + ↵) _{⇡ e}F_int = ˜ n X i=1 e i i = ˜

= arg min_kF_int ˜ kX

e F_int = P F_int = ✓ ˜ ⇣˜T_{X ˜}⌘ 1 ˜T_X◆_F int 4 System Approximation

Constraining the displacement in a low-dimensional space does not provide a significant computational gain, even if the systems to be solved are of smaller dimension. This is because the material of study is nonlinear and history-dependent, and its sti↵ness varies not only in di↵erent areas of the material but also with time. This requires to evaluate the sti↵ness everywhere in the material and this at each time step of the simulation. This means that the numerical complexity remains despite the simplification on the displacement. Hence, to decrease the numerical complexity, the domain itself need to be approximated. Several authors have looked into that. Notable contributions include the Hyperreduction method [15], the missing point estimation [16], or system approximation [17]. Those methods share the idea that the material properties will be evaluated only at a small set of points or elements within the material domain. They di↵er in the way of selecting those points and in the treatment of that reduced information. In this paper, we will use the ”gappy” method, very much like in [17, 18].

4.1 Gappy Method

The internal forces generated by the reduced displacement Fint( ↵) will be evaluated only in a small

subset of the degrees of freedom _{I of the domain ⌦. All the elements in contact with those degrees} of freedom have to be considered. We refer to those as the controlled elements. The internal forces will then be reconstructed by writing the internal forces as a linear combination of a few basis vectors themselves (just like it was made for the displacement).

Fint( ↵)⇡ n_Xgap 1 i i= , (29) whereh 1,· · · , ngap i

= is the forces basis of size ngap and the associated scalar coefficients.

(a) Original structure (b) Example of a surrogate structure

Figure 5: Example of a surrogate structure. The sti↵ness of the structure is evaluated on controlled elements only, while the other ones are just like ghosts

The coefficients of the expansion are found so that to minimise the norm of the di↵erence between the linear expansion and the nonlinear term over the subset_I:

argmin

? kFint( ↵)

?

kP, (30)

9

Constraining the displacement in a low-dimensional space does not provide a significant computational gain, even if the systems to be solved are of smaller dimension. This is because the material of study is nonlinear and history-dependent, and its sti↵ness varies not only in di↵erent areas of the material but also with time. This requires to evaluate the sti↵ness everywhere in the material and this at each time step of the simulation. This means that the numerical complexity remains despite the simplification on the displacement. Hence, to decrease the numerical complexity, the domain itself need to be approximated. Several authors have looked into that. Notable contributions include the Hyperreduction method [15], the missing point estimation [16], or system approximation [17]. Those methods share the idea that the material properties will be evaluated only at a small set of points or elements within the material domain. They di↵er in the way of selecting those points and in the treatment of that reduced information. In this paper, we will use the ”gappy” method, very much like in [17, 18].

4.1 Gappy Method

The internal forces generated by the reduced displacement Fint( ↵) will be evaluated only in a small

subset of the degrees of freedom _{I of the domain ⌦. All the elements in contact with those degrees} of freedom have to be considered. We refer to those as the controlled elements. The internal forces will then be reconstructed by writing the internal forces as a linear combination of a few basis vectors themselves (just like it was made for the displacement).

Fint( ↵)⇡ n_Xgap 1 i i= , (29) where h 1,· · · , ngap i

= is the forces basis of size ngap and the associated scalar coefficients.

(a) Original structure (b) Example of a surrogate structure

Figure 5: Example of a surrogate structure. The sti↵ness of the structure is evaluated on controlled elements only, while the other ones are just like ghosts

The coefficients of the expansion are found so that to minimise the norm of the di↵erence between the linear expansion and the nonlinear term over the subset_I:

argmin

? kFint( ↵)

?

kP, (30)

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• Basis func$ons: consistent approach

§  Basis obtained by standard empirical POD for

→  In prac$ce, solve the snapshots a second $me using standard Galerkin-‐ POD and use the residuals of the Newton solver

§  Truncate the POD expansion at an order such that displacement and stress errors are of the same order of magnitude

• Interpola$on points minimise reconstruc$on error (EIM [Barrault ‘04], MPE [Astrid ‘08])

Choosing the parameters of the Gappy approx

F_int (↵(t) + ↵)

Mode 1

Mode 2

Figure 9: Regions of interest selected by the system approximation procedure. Those regions (circled in the Figure) are matching the areas of higher displacement found in the POD bases. This is intuitively good, since those elements have to give enough information to be able to reconstruct the internal forces over the entire domain. Those are the elements whose behaviour vary the most when changing the loading path (which is the parameter of the reduced model), hence containing the core information necessary to build up an accurate reconstruction.

5.4 Numerical results

In this section, we will test the performance of the method by comparing the relative error between the ”truth“ solution of the RVE problem, which is the solution obtained when using the full order model, and the reduced model. We will focus on the simulation of the RVE problem only, and not consider the complete computational homogenisation framework (the macroscale problem will not be solved). For the first test, the load path is set using the following e↵ective strain: ✏M(t) = _Tt .

 1 1

1 1 . Note that this case is not in the snapshot. First, the relative error is plotted with various numbers of POD basis vectors. See Figure 10.

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• Snapshot:

Test case

1 2 3 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

100 QR sampling in the macro strain space

SVD

The basis is selected using the POD method explained in section 3.2. The first few modes are

displayed in Figure 7. We will test the results using various number of basis vectors. Procedures to select the optimal number of modes according to a robust cross-validation procedures can be found in [11].

(a) Mode 1 (b) Mode 2

(c) Mode 3

Figure 7: First 3 modes obtained through the POD. The damage localises between pairs of inclusions.

5.3 System approximation

We follow the procedure described in 4. The basis is extracted from the same snapshot space

as used for the displacement basis . The set of controlled elements is selected using the DEIM

[18]. The amount of vectors in the basis is chosen so that the system approximation does not

increase the global error of the reduced order model. The error ⌫totbetween the exact solution and

the reduced model solution with system approximation can be decomposed in the following way (with

uex(t) the exact solution, ur(t; ), the reduced order solution without the system approximation using

the dynamic basis , and ur,sa(t; , ) the complete reduced order model with system approximation

12

5.2 POD basis

(c) Mode 3

12

5.2 POD basis

(c) Mode 3

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• Valida$on for one par$cular loading

Speed-‐up

2 Speed-‐up 15

Last $me step before localisa$on:

15 SU 10

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Op$mal sta$c vs kinema$c surrogate sizes

Figure 10: Relative error when using the reduced model without system approximation. As the number of POD basis vectors increases, the error decreases. Using more than 4 basis vectors does not have a strong e↵ect. The error reaches a threshold. That threshold corresponds to the projection error of the exact solution onto the snapshot space.

2 4 6 8 10 0 20 40 60 80 100 10⌥5 10⌥4 10⌥3 10⌥2 10⌥1 100 Rel ati ve e rr or in energy no rm Number of vector_s in the stati

c basis Number of vector

s in the displacement

basis

Figure 11: Evolution of the error varying the number of displacement and static basis vectors.

Several remarks can be made:

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Automa$sed snapshot selec$on

✏11

✏12

Ini$alisa$on (propor$onal loading)

Load history that maximises the ROM error (obtained by numerical op$misa$on)

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• Numerical solu$on of RVE problems avoids assump$ons on microscale ﬁelds, but too expensive

• Reduced order modelling permits to alleviate this problem, but

providing error-‐controlled approxima$ons that seamlessly span from full-‐FE solu$on to highly-‐reduced RVE problems

• Open-‐ques$on: mul$scale modelling and reduced order modelling do the same thing: look for and use invariances in solu$on sets. Is there a more formal way to couple them?

Reduced order modelling: towards tractable computational homogenisation schemes

Reduced order modelling: towards tractable

computa5onal homogenisa5on schemes

Team in Cardiﬀ: main research topics

Team in Cardiﬀ: main research topics

Toolkit 1: advanced discre$sa$on methods

Toolkit 2: mul$level methods

Toolkit 2: mul$level methods

Mul$scale modelling

Examples

Mathema$cal formula$on

Classical homogenisa$on in linearised elas$city

Classical homogenisa$on in linearised elas$city

Computa$onal homogenisa$on

Virtual charts

ROM-­‐based Virtual charts

ROM-­‐based Virtual charts

Reduced order model

Galerkin-­‐POD with QRandom sampling

Key issue 1: choice of samples

Convergence es$mate

Alterna$ve methods for snapshot selec$on

Key issue 2: eﬃcient assembly of ROM

Goal-­‐oriented error es$ma$on

Error bounding by duality

Local error es$ma$on

Schema$c of duality-­‐based ROM

Results

Reduced basis sampling technique

GO hybrid reduced basis approach

GO hybrid reduced basis approach

Illustra$on for compliant problem

Nonlinear comput. homogenisa$on by FE

Damageable RVE

Realisation 1

Realisation 2

Particle

distribution

Projection of

the material

properties

Figure 5: Two realisations of the random distribution of the material properties. The inclusions

are spherical, and their projection onto the lattice structure is represented in black. The grey bars

correspond to the matrix of the particulate composite, while the light grey ones define the weak

interface between aggregates and matrix.

In order to describe the randomness of the model formally, we introduce the probability space

H = (⇥, F, P ) for the random distribution of the material properties of the lattice network. ⇥ is the

ensemble of outcomes of the random generation,

F is the corresponding Sigma-algebra of subsets of ⇥

and P is the associated probability measure. Any random distribution is characterised by distribution

function (26), a fixed maximum inclusion diameter D

, a fixed phase ratio, fixed material properties

for each of the phases, and a given lattice network.

Two realisations of the random distribution of material properties are illustrated in figure 5.

2.4

Discretisation

We now present briefly the discretisation technique of the lattice balance equations (15). The

displace-ment of a point of the neutral fibre of beam (b) is approximated using a unique finite eledisplace-ment, of first

order degree for the normal displacement, and third order degree for the deflection:

✓

v

w

◆

(⇠) =

(⇠)

(⇠)

!

0

B

B

B

B

B

B

@

v

(0)

w

(0)

✓

ROM-‐based Virtual charts

ROM-‐based Virtual charts

Galerkin-‐POD with QRandom sampling

Goal-‐oriented error es$ma$on

Schema$c of duality-‐based ROM

_{F is the corresponding Sigma-algebra of subsets of ⇥}