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 (I 2 SC) 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: � � L u ( x ) = T � u ( x ) , u ( y ) , x , y � − T � u ( y ) , u ( x ) , y , x � � d y • Connection to discrete models (MD): � � � T � u k , u j , x k , x j � − ˆ ˆ L u k ∼ T � u j , u k , x j , x k � j • Connection to local continuum models (PDEs): � � lim ω ǫ ( | y − x | )( u ( y ) − u ( x )) dy = ∆ y ( δ | y − x ))( u ( y ) − u ( x )) dy = ∆ u ( x ) ǫ → 0 • Connection to graphical models (graph Laplacian, kernel estimation, diffusion map): � L u k ∼ ω jk ( u j − u k ) j 9 / 26
An illustration: bond-based peridynamics Force balance for a continuum of (linear/isotropic) Hookean springs: −L δ u δ = b in Ω . � y − x � ω δ ( | y − x | ) y − x �� L δ u ( x ) = | y − x | 2 · � u ( y ) − u ( x ) d y . | y − x | 2 Ω ∪ Ω δ u δ ( x ) : displacement at x ; Spring force for y ∈ B δ ( x ) ; y − x : bond direction; δ : nonlocal horizon; Ω ω δ ( | r | ) : nonnegative, Ω δ u δ = 0 supported in B δ ( 0 ) . Volumetric constraint: u δ = 0 in Ω δ = { x ∈ Ω c , d ( x , ∂ Ω) < δ } , or a subset of Ω δ . 10 / 26
Reformulation Problem: find u δ , −L δ u δ = b in Ω , u δ =0 in Ω δ . Ω � � D ∗ �� Rewrite L δ as D ω δ : Ω δ u δ = 0 ω δ ( | y − x | ) y − x � � � D ∗ ( u ) �� | y − x | 2 D ∗ ( u )( x , y ) d y . D ω δ ( x ) = y − x D ∗ ( u )( x , y ) = � � | 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: a δ ( u , v ) = < b , v >, ∀ v ∈ V δ (a nonlocal function space) �� � ω δ ( D ∗ u )( D ∗ v ) d y d x , < b , v > = where a δ ( u , v ) = bv d x , 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 � �� ω δ | D ∗ u | 2 ( ∇ u ) T K ∇ u Local energy ⇔ Nonlocal energy Local flux ⇔ Nonlocal flux Sobolev space H 1 (Ω) ⇔ Nonlocal function space V δ � � �� u D ( D ∗ v ) − v D ( D ∗ u ) = 0 u ∆ v − v ∆ u = u ∂ n v − v ∂ n u ⇔ Ω ∂ Ω Local balance (PDE) ⇔ Nonlocal balance (PD) −∇ · ( K ∇ u ) = f ⇔ − D · ( ω δ D ∗ 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 u δ ∈ V δ Local PDE limit u 0 ∈ V 0 u δ = 0 ← Volumetric constraint Well-posed with Ω Ω Ω δ ← → ∂ Ω - L δ u δ = b a unique solution - L 0 u 0 = b Boundary condition → u 0 = 0 • As δ → 0, we have u δ → u 0 in L 2 under minimal regularity (Mengesha-Du) . ⋆ u 0 solves the Navier system of linear elasticity with a Poisson ratio 1 / 4. ⋆ Nonlocal solutions { u δ } δ> 0 may be less regular than the local limit u 0 . ⋆ Results hold for suitable nonlocal kernels ω δ ( 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. (Parks et al, Sandia, Has PD avoided all of these issues? 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 u δ Local PDE limit u 0 u δ = 0 Ω Ω Ω δ ∂ Ω h - L δ u δ = b - L 0 u 0 = b u 0 = 0 ⇒ u h u h • Numerical solutions: h → 0 = δ → u δ (nonlocal) 0 → u 0 (local) 0 (verification/benchmark) ? • With u δ → u 0 as δ → 0, do we have u h δ → u h 15 / 26
Discretization of nonlocal PD and local limit • An abstract depiction of convergence issues for parametrized problem u h u h Discrete Discrete δ 0 Nonlocal Local h = 0 h → 0 h → 0 Continuum Continuum δ → 0 u δ u 0 Nonlocal h = 0 PDE δ = h = 0 Q : lim δ → 0 u h δ = u h 0 ? lim δ → 0 , h → 0 u h δ = u 0 ? (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. u h u h Discrete Discrete δ 0 δ → 0 Local δ = 0 Nonlocal sparse h → 0 h → 0 h → 0 δ → 0 dense Continuum Continuum δ → 0 u δ u 0 Nonlocal h = 0 PDE δ = h = 0 Q: What schemes are AC? Examples in Tian-Du 2013 SINUM, general theory in Tian-Du 2014 SINUM 18 / 26
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