Nonlocal models as effective bridges in multiscale modeling Qiang - - PowerPoint PPT Presentation

Nonlocal models as effective bridges in multiscale modeling

Qiang Du

• Dept. of Appl. Phys.& Appl. Math, & Data Science Institute,

Columbia Univ

1 / 26

Predictive multiscale materials modeling

Some past adventures and reflections: NSF Project (MATH+STAT+CS+PHYS) 2004 — CAMLET: A Combined Ab-initio Manifold LEarning Toolbox for Nanostructure Simulatiions. Intelligent and Informative Scientific Computing (I2SC) 2006 ⇒ Simulation + Data + Mining + Learning. ... 2015: UQ Projects with DARPA (ORNL/Columbia/FSU/Warwick/RWTH/TAMU), NSF DMREF (Columbia), and AFRL STTR (with founder/CEO of Materials Genome Inc)

2 / 26

Predictive multiscale materials modeling

Now with Computational-Science-Engineering/BigDATA/MGI becoming big business, let us talk about: risk control.

3 / 26

Nonlocality is ubiquitous

Nonlocality: generic feature of multiscale modeling/model reduction. Nonlocal (integral) models & simulations: long history and rich literature. We discuss: a systematic (axiomatic) mathematical framework and its application. Highlight: reduce risk with robust algorithms for problems having varying scales. Collaborators: X. Tian, T. Mengesha, M. Gunzburger, R. Lehoucq, K. Zhou,

• L. Ju, J. Kamm, M. Parks, L.Tian, X.Zhao, H.Tian, A.Tartakovsky, Z.Huang,

P .Lefloch, Y.Tao, J.Yang, Z.Zhou Supported in part by NSF-DMS and AFOSR-MURI

4 / 26

Classical, local balance laws

In classical continuum mechanics, conservation laws are often given as PDEs, e.g., the classical equation of motion: ρ ¨ y(x, t) = ∇ · σ + b , where y: position of x; F: deformation gradient; σ = ˆ σ(F): constitutive model. Limitation: validity is in question near materials defects such as cracks. Problems with classical approaches for cracks:

• Tend to get different results on different mesh
• Do not reflect realistic crack-tip processes
• Difficult to apply to complex crack trajectories
• Destroy the accuracy/convergence properties.

Remedy? multiscale modeling and simulations, eg, atomistic model near cracks and elasticity away from cracks ⇒ marrying dis- similar equations, a challenging and ongoing investigation.

5 / 26

Peridynamics

A nonlocal alternative to local mechanics by Silling 2001 (Belytschko prize 2015). Same equation on and off material defects/singularities, with no spatial derivatives. (Silling) to unify the mechanics of continuous and discontinuous media.

6 / 26

Peridynamics

• Peridynamics (PD) uses partial-integral equations and has found many applications.

(Silling) WIthout spatial derivatives, cracks (singularities) are allowed as part of the solution.

7 / 26

PD based simulations

There have been significant code development efforts (PDLAMMPS, PERIDIGM...)

• Singularities (cracks/fractures) may make peridynamics (PD) closer to reality, but

increased complexities also demand better mathematical theory, efficient and robust algorithms and careful validation/verification (VV).

• Yet, studies of nonlocal models (such as PD) have not shared a path parallel

to that of local continuum models/PDEs (Newton’s calculus), until recent years.

8 / 26

Nonlocal continuum models

Models in terms of nonlocal/integral continuum operator: Lu(x) = Tu(x), u(y), x, y − Tu(y), u(x), y, x

• dy
• Connection to discrete models (MD):

Luk ∼

• j
• ˆ

Tuk, uj, xk, xj − ˆ Tuj, uk, xj, xk

• Connection to local continuum models (PDEs):

lim

ǫ→0

• ωǫ(|y − x|)(u(y) − u(x))dy =
• ∆y(δ|y − x))(u(y) − u(x))dy = ∆u(x)
• Connection to graphical models (graph Laplacian, kernel estimation, diffusion map):

Luk ∼

• j

ωjk(uj − uk)

9 / 26

An illustration: bond-based peridynamics

Force balance for a continuum of (linear/isotropic) Hookean springs: −Lδuδ = b in Ω. Lδu(x) =

• Ω∪Ωδ

ωδ(|y − x|) y − x |y − x|2 y − x |y − x|2 ·

• u(y) − u(x)
• dy .

uδ(x): displacement at x; Spring force for y ∈ Bδ(x); y − x: bond direction; δ : nonlocal horizon; ωδ(|r|): nonnegative, supported in Bδ(0). Ω Ωδ uδ = 0 Volumetric constraint: uδ = 0 in Ωδ = {x ∈ Ωc, d(x, ∂Ω) < δ}, or a subset of Ωδ.

10 / 26

Reformulation

Problem: find uδ, −Lδuδ = b in Ω, uδ=0 in Ωδ. Rewrite Lδ as D

• ωδ
• D∗

: Ω Ωδ uδ = 0 D

• ωδ
• D∗(u)
• (x) =
• ωδ(|y − x|) y − x

|y − x|2 D∗(u)(x, y) dy . D∗(u)(x, y) = y − x |y − x|2 ·

• u(y) − u(x)
• linear nonlocal volumetric strain.

D: dual/adjoint operator of D∗, < D(ϕ), u >=< ϕ, D∗(u) >∗ , ∀ ϕ, u. Principle of virtual work: aδ(u, v) =< b, v >, ∀ v ∈ Vδ (a nonlocal function space) where aδ(u, v) =

• ωδ (D∗u)(D∗v)dydx , < b, v >=
• bvdx ,

D∗, D, and integral identities: part of nonlocal vector calculus. Du-Gunzburger-Lehoucq-Zhou 2013 M3AS, D-G-L-Z 2012 SIAM Review Well-posedness and properties: nonlocal calculus of variations D-G-L-Z 2012, 2013; Mengesha-Du 2013, 2014, 2015, Tian-Du 2015, ...

11 / 26

Nonlocal vector calculus

Newton’s vector calculus ⇔ Nonlocal vector calculus Differential operators ⇔ Nonlocal operators Local energy

• (∇u)TK∇u

⇔ Nonlocal energy

• ωδ|D∗u|2

Local flux ⇔ Nonlocal flux Sobolev space H1(Ω) ⇔ Nonlocal function space Vδ

• Ω

u∆v − v∆u =

• ∂Ω

u∂nv − v∂nu ⇔

• uD(D∗v) − vD(D∗u) = 0

Local balance (PDE) ⇔ Nonlocal balance (PD) −∇ · (K∇u) = f ⇔ −D · (ωδD∗u) = f Boundary conditions ⇔ Volumetric constraints Goal: systematic/axiomatic framework, mimicing classical/local calculus for PDEs.

12 / 26

Nonlocal problems and local limit

Nonlocal problem uδ ∈ Vδ Local PDE limit u0 ∈ V0 Ω Ωδ

• Lδuδ= b

uδ= 0 Well-posed with ← → a unique solution ← Volumetric constraint Boundary condition → Ω ∂Ω

• L0u0= b

u0= 0

• As δ → 0, we have uδ → u0 in L2 under minimal regularity (Mengesha-Du).

⋆ u0 solves the Navier system of linear elasticity with a Poisson ratio 1/4. ⋆ Nonlocal solutions {uδ}δ>0 may be less regular than the local limit u0. ⋆ Results hold for suitable nonlocal kernels ωδ(r) = ω(r/δ)δ−2−d. Bottomline: consistency on the continuum level with local models (if valid).

13 / 26

Numerical solution of PD

Quadrature approximations lead to (very popular) mesh-free/discrete-particle methods. Variational forms lead to FEM and other schemes, similar to those for local models. Recall those limitations quoted earlier on the traditional ways for simulating cracks:

• Tend to get different results when changing mesh
• Do not reflect realistic crack-tip processes
• Difficult to apply to complex crack trajectories
• Destroy the accuracy/convergence properties.

Has PD avoided all of these issues? (Parks et al, Sandia, PeriDigm Manual) Despite many successful simulations, there were also plenty complaints ...... Chief among them: inconsistency with ABACUS for simple bench-mark tests.

14 / 26

Numerical solution

Nonlocal problem uδ Local PDE limit u0 Ω Ωδ

• Lδuδ= b

uδ= 0 h Ω ∂Ω

• L0u0= b

u0= 0

• Numerical solutions: h → 0 =

⇒ uh

δ → uδ (nonlocal)

uh

0 → u0 (local)

• With uδ → u0 as δ → 0, do we have uh

δ → uh 0 (verification/benchmark) ? 15 / 26

Discretization of nonlocal PD and local limit

• An abstract depiction of convergence issues for parametrized problem

uh

δ

uh uδ u0

Discrete Nonlocal Continuum Nonlocal h = 0 Discrete Local h = 0 Continuum PDE δ = h = 0 h → 0 δ → 0 h → 0

Q: limδ→0 uh

δ = uh 0 ? limδ→0,h→0 uh δ = u0 ? (benchmark against known local

solutions, when they are valid, is often the 1st step of code verification).

16 / 26

Effective/robust solution of parametrized problems

A popular subject of numerical analysis associated with applications in many fields. ◮ Frameworks based on perturbation/continuation (Brezzi-Rappaz-Raviart,...). ◮ Many existing studies on the effective discretization in limiting regimes:

• Asymptotic preserving schemes for NLS in semiclassical limit

(Jin, Filbet, Degond, Bao, Besse, Carles, Mehats ...)

• Numerical discretization of radiative transfer in diffusive limit

(Degond, Carrillo, Lafitte, Guermond-Kanschat, Pareschi-Russo, ...)

• Locking-free finite element methods for elasticity models

(Arnold-Brezzi. Girault-Raviart, Chapelle-Stenberg, Reddy, Fortin, ... )

• Other examples include convection-dominated problems and other singular

perturbation problems, sharp-interface limit of diffuse interface models, ... ... ◮ Distinct and new: solving nonlocal models and local limits (Tian-Du 2014).

17 / 26

Asymptotically compatible (AC) schemes

• AC schemes (Tian-Du): schemes that are convergent as h → 0 to nonlocal

models with fixed δ, and as δ → 0, h → 0 to the corresponding local limit.

uh

δ

uh uδ u0

Discrete Nonlocal Continuum Nonlocal h = 0 Discrete Local δ = 0 Continuum PDE δ = h = 0 δ → 0 h → 0 δ → 0 h → 0 δ → h → 0 sparse dense Q: What schemes are AC? Examples in Tian-Du 2013 SINUM, general theory in Tian-Du 2014 SINUM

18 / 26

Asymptotically compatible schemes

Unfortunately, some most popular schemes are not! (Tian-Du 2013) ◮ Midpoint quadrature approximation of integrals is not AC. ◮ Finite element with piecewise constant functions (an easy choice) is not AC. Keeping δ/h as a constant (for sparsity), they may converge, but to a wrong local limit!

uh

δ

uh uδ u0

• h → 0

δ → 0 h → 0

• h → 0, δ → 0

They over-estimate elastic moduli by constant factors (depending on δ/h) as h → 0, incompatible to the correct local limit. Unfortunately, early algorithmic development by the community has been on such schemes, causing serious issues for VV.

19 / 26

Asymptotically compatible schemes

How to do better? Tian-Du 2014 SINUM provided an abstract framework and specified conditions for AC schemes. In particular, for nonlocal PD systems in multi-dimensions: Theorem (Tian-Du 2014): any continuous or discontinuous conforming FEM containing all continuous linear elements is AC, thus is good for both nonlocal and local regimes.

uh

δ

uh uδ u0

Discrete Nonlocal Continuum Nonlocal h = 0 Discrete Local h = 0 Continuum PDE δ = h = 0 δ → 0 h → 0 δ → 0 h → 0 δ → 0 h → 0 sparse dense ⇒ AC if containing C0 pw linears. For pw constants, conditional AC if h/δ → 0.

20 / 26

AC: specialized to nonlocal problems

• This exercise exposed risks associated with some popular algorithms for nonlocal

models (failing simple benchmark tests due to the lack of AC property).

• High order alternatives are AC, robust/low-risk to the change of spatial scales (δ).

Previously it is thought impossible to have convergent schemes without h/δ → 0. These results are impacting the community’s view on how to design ”nonlocal” codes.

• The framework is general (for parametrized problems), works for nonlocal systems.

Minimal assumptions on mesh/particle distribution (key: dense approximation), a total departure from classical notions (polynomial reproducing, moment vanishing,...)

• Applications to multidimensional nonlocal systems require extensions of works of

Bourgain-Brezis-Mironescu 2001, Ponce 2004 to characterize nonlocal spaces. Motivating further generalizations and applications: L2 to Lp (Mengesha-Du 2015 Nonlinearity) ⇒ nonlinear elasticity; Local to nonlocal limit (Tian-Du 2015 SINUM) ⇒ DG; Low (1st) to high order nonlocal operators(Tian-Du 2015) ⇒ beam/plate;

21 / 26

Nonlocal models in concurrent multiscale simulations

22 / 26

Nonlocal models for concurrent multiscale simulations

Coarse grained PD (Silling) and PD with variable horizon (Littlewood, Seleson, Silling) (Silling)

23 / 26

Concurrent multiscale simulations

Popular concurrent multiscale coupling: CE+MD, CE+QC+MD,... CE + Nonlocal blending + MD To better understand the nonlocal effect in domain composition:

Local (CE) + Nonlocal (with overlap) Without overlap (Tian-Du, Du-Tao-Tian)

Local Local

Key: on ∂Ω, local H1 solution has a trace, to be matched with that of nonlocal solution. Asymptotically compatibility provides the much needed robustness and adaptivity.

24 / 26

Summary

• Nonlocality is ubiquitous.
• We attempt to establish a systematic (and perhaps axiomatic)

mathematical framework for nonlocal models/discretization.

• It contains nonlocal vector calculus, nonlocal calculus of

variations, and asymptotically compatible discretization.

• It is informing (PD) community the need for low-risk codes

in nonlocal (multiscale) modeling, making impact on V/V.

• It may have broad applications, given the connection to fractional calculus, discrete

exterior calculus, graph calculus, Levy processes, diffusion maps, combinatorial Hodge theory, Mori-Zwanzig, Quasi-continuum, SPH, RKPM, .... Thanks to: Xiaochuan Tian, T. Mengesha, M. Gunzburger, R. Lehoucq, K. Zhou, ,,, Supported in part by NSF-DMS, AFOSR-MURI

25 / 26

SIAM MS16

Over 140 minisymposia. All areas of SIAG-MS, including themes close to materials genome Deadline: 12/15/2015 Contributed Lecture, Poster and Minisymposium Presentation Abstracts Travel support to students Join us in Philly! Contact: meetings@siam.org

• Q. Du, D. Srolovitz

26 / 26