# Schedule of the Workshop on Non-local Material Models and Concurrent Multiscale Methods

## Monday, April 3

10:15 - 10:50 |
Registration & Welcome coffee |

10:50 - 10:55 |
Opening remarks |

10:55 - 11:45 |
Ricardo Nochetto: Two-scale FEMs for non-variational elliptic PDEs and pointwise convergence rates |

11:45 - 12:35 |
Wei Zhao: Localized radial basis functions-based pseudo-spectral method (LRBF-PS) for nonlocal diffusion problems |

12:35 - 14:20 |
Lunch break |

14:20 - 15:10 |
Xiaochuan Tian: Asymptotically Compatible Discretization of Nonlocal Variational Problems |

15:10 - 16:00 |
Markus Schmuck: Porous media approximation of phase-field equations for interfacial dynamics |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:20 |
Jiun-Shyan Chen: Fracture to Damage Multiscale Mechanics and Modeling of Brittle Materials |

afterwards |
Reception |

## Tuesday, April 4

09:30 - 10:20 |
John Foster: Finite deformation constitutive models and mechanics of peridynamic mixtures |

10:20 - 10:50 |
Group photo and coffee break |

10:50 - 11:40 |
Pablo Seleson: Concurrent Multiscale Methods for Local/Nonlocal Coupling |

11:40 - 12:30 |
Jeremy Trageser: Anisotropic Linearized Peridynamic Models |

12:30 - 14:20 |
Lunch break |

14:20 - 15:10 |
Guanglian Li: Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d |

15:10 - 16:00 |
Mira Schedensack: Relaxing the CFL condition for the wave equation on adaptive meshes |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:20 |
Anter El-Azab: Mesoscale crystal plasticity based on continuum dislocation dynamics: mathematical formalism and numerical solution |

## Wednesday, April 5

09:00 - 09:50 |
Wing Kam Liu: Self-consistent clustering analysis for data-driven design of multiscale material systems |

09:50 - 10:40 |
Jiang Yang: Asymptotically compatible discretization of nonlocal models: quadrature-based finite difference and Fourier spectral methods |

10:40 - 11:10 |
Coffee break |

12:00 - 18:00 |
Excursion on the Rhine |

## Thursday, April 6

09:30 - 10:20 |
Michael Ortiz: Deformation microstructure, relaxation and numerical approximation |

10:20 - 10:50 |
Coffee break |

10:50 - 11:40 |
Marta D'Elia: A coupling strategy for nonlocal and local models with applications to static peridynamics and classical elasticity |

11:40 - 12:30 |
Marc Alexander Schweitzer: tba |

12:30 - 14:20 |
Lunch break |

14:20 - 15:10 |
Juan Carlos De los Reyes: Bilevel learning approaches in variational image processing: From local to non-local models |

15:10 - 16:00 |
Panayot Vassilevski: Spectral upscaling for graph Laplacian problems with application to reservoir simulation |

16:00 - 16:30 |
Tea and cake |

18:30 - |
Conference dinner in the Restaurant "Brauhaus Bönnsch" (Sterntorbrücke 4) |

## Friday, April 7

09:30 - 10:20 |
Soeren Bartels: Numerical solution of a BV regularized damage evolution model |

10:20 - 10:50 |
Coffee break |

10:50 - 11:40 |
Benjamin Stamm: A perturbation-method-based post-processing of planewave approximations for Density Functional Theory (DFT) models |

11:40 - 12:30 |
Dietmar Gallistl: On quasi-local discrete models in numerical homogenization |

12:30 |
End of the workshop |

# Abstracts

## Soeren Bartels: Numerical solution of a BV regularized damage evolution model

Functions of bounded variation provide an attractive framework to model mechanical effects that lead to fracture and damage. Their numerical approximation is difficult due to nondifferentiability of related contributions to energy functionals and limited regularity properties including jumps across lower dimensional subsets. In the talk we first discuss the finite element discretization and iterative solution of a model problem from image processing. In particular, we discuss a priori and a posteriori error estimates and devise a variant of the alternating direction method of multipliers with variable step sizes for the efficient numerical solution. The developed techniques are then applied to a BV-regularized model for damage evolution which is defined via a nonconvex energy functional and a degree-one homogeneous dissipation potential. The talk is based on joint work with Marijo Milicevic (University of Freiburg) and Marita Thomas (WIAS Berlin).

## Jiun-Shyan Chen: Fracture to Damage Multiscale Mechanics and Modeling of Brittle Materials

The failure processes in the materials exhibit distinct characteristics depending on the material ductility, the loading rate, and the environmental conditions. The mathematical models and the associated numerical methods for describing the material failure processes can be classified as the discrete description based on fracture mechanics and the continuum phenomenological description based on damage mechanics. This work first discusses how damage mechanics based models can be formulated as the homogenization of fracture models. The challenges in the numerical approximation and discretization of failure modeling based on fracture mechanics and damage mechanics will then be addressed. Model order reduction techniques for enriched reproducing kernel meshfree methods are introduced for analysis of problems with singularities. The employment of integrated singular basis function method (ISBFM), in conjunction with the selection of harmonic near-tip asymptotic basis functions, leads to a Galerkin formulation in which the non-smooth near-tip basis functions appear only on the boundaries away from the singularity point. The size effect in bridging fracture mechanics to damage mechanics is identified, and "implicit gradient" approach and scaling law are introduced to remedy the associated mesh dependency.

## Marta D'Elia: A coupling strategy for nonlocal and local models with applications to static peridynamics and classical elasticity

The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local Partial Differential Equation (PDE) models motivates and drives the interest in this type of simulations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. In particular, a complete nonlocal simulation remains computationally untenable for many science and engineering applications. As a result, it is important to develop local-to-nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model.

We develop and analyze an optimization-based method for the coupling of nonlocal and local problems in the context of nonlocal elasticity. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We prove that the resulting optimization problem is well-posed and discuss its implementation using Sandia's agile software components toolkit.

Numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method; these numerical tests provide the groundwork for the development of efficient and effective engineering analysis tools. As an application, we present results for the coupling of static peridynamics and classical elasticity.

## Juan Carlos De los Reyes: Bilevel learning approaches in variational image processing: From local to non-local models

In order to determine the noise model in corrupted images, we consider a bilevel optimization approach in function space with the variational image denoising models as constraints. In the flavour of supervised machine learning, the approach presupposes the existence of a training set of clean and noisy images. The problems are treated as mathematical programs with variational inequality constraints and differentiability properties of the solution operators are investigated. Optimality conditions in form of Karush-Kuhn-Tucker systems are derived and, based on the adjoint information, fast Newton type algorithms are proposed for the numerical solution of the problems. We consider different cost functionals (L2, TV) and different local regularizers (TV, TGV, ICTV), and compare their performance for the set of images at hand. We will also discuss the extension of the approach to cope with non-local variational models and the main difficulties and challenges presented. A comparison of local and non-local denoising models will be carried out by means of test images.

## Anter El-Azab: Mesoscale crystal plasticity based on continuum dislocation dynamics: mathematical formalism and numerical solution

Prediction of the self-organization of dislocations (line defects) in metal crystals has captured the interest of theoretical physicists, mathematicians, mechanics researchers, and metallurgists for over half a century, for its critical importance to understanding plastic strength and failure of metals. Recently, a simulation based approach named "dislocation dynamics" was introduced to solve this problem but quickly proved that it is computationally non-viable. Alternate approached for developing density-based mathematical models of dislocations then appeared which use the principles of statistical mechanics and non-equilibrium thermodynamics to developing physically sound and tractable mathematical models of dislocation dynamics. We will present the first working model in this regard. The model boils down to a large system transport-reaction equations of the div-curl type that govern the evolution of the dislocation system in crystals, which are augmented by the equations of linear elasticity. Because of the long range nature of stress of dislocations, the system represents a local formalism of the non-local dislocation dynamics problem. Other aspects of non-locality, featured in reactions-at-a-range and in two-point correlations with lengths are also explained. A novel Least Squares variational formalism has been used to solve the overall problem using the finite element method and a physically motivated meshing scheme aiming to accurately capture the planar dislocation transport in 3D space. The effectiveness of the modeling and numerical approach will be demonstrated by predicting various dislocation patterns observed experimentally for the first time in the history of dislocation theory and crystal plasticity theory. This work is conducted in collaboration with Shengxu Xia.

## John Foster: Finite deformation constitutive models and mechanics of peridynamic mixtures

The conservation of momentum equations for superimposed interacting materials, i.e. mixtures, can be rigorously derived using an extension of Hamilton’s principle whereby constraint equations are appended to the action functional and enforced through Lagrange multipliers. The key constraint equations utilized in the derivations are the volume fraction constraint and the material form of conservation of mass. When considering finite deformations, the volume fraction constraint must be enforced in the deformed configuration which might require pull-back operations to the reference configuration. Pull-back operations have not been well-developed in the context of peridynamic mechanics. Likewise, the material form of conservation of mass in the classical theory utilizes the determinant of the deformation gradient, the Jacobian determinant, to described the volume scaling between the reference and deformed configurations. In peridynamic theory, the displacement field can be discontinuous and therefore the deformation gradient is potentially undefined. Additionally, in the modeling of solid materials, the peridynamic pressure at a material point is considered to be a function of the totality of deformation within a nonlocal region of the point. This implies that the material points density is influenced nonlocally as well, i.e. a nonlocal conservation of mass. Therefore, we propose a "peridynamic Jacobian determinant" that represents the volume scaling of the nonlocal interaction region between the reference and deformed configurations. In this work we present an efficient tensor measure to compute this peridynamic Jacobian determinate and it’s Frechet derivative. Additionally, we present nonlocal tensor analogues of the left- and right-Cauchy Green deformation tensors and show how they can be used in pull-back operations and constitutive modeling of peridynamic materials. Finally, we develop finite deformation conservation of momentum equations for a two-specie peridynamic mixture that is in analogy with classical poromechanics.

## Dietmar Gallistl: On quasi-local discrete models in numerical homogenization

The talk discusses a re-interpretation of existing multiscale methods by means of a discrete integral operator acting on standard finite element spaces. The exponential decay of the involved integral kernel motivates the use of a diagonal approximation and, hence, a localized piecewise constant coefficient. This new point of view is very useful for numerical stochastic homogenization. An a priori error analysis is presented and illustrated in numerical experiments.

## Guanglian Li: Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d

This is joint work with Daniel Peterseim and Mira Schedensack. We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on precomputed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional inter-element communication.

## Wing Kam Liu: Self-consistent clustering analysis for data-driven design of multiscale material systems

The advent of advanced processing and manufacturing techniques provides unparalleled freedom to design new material classes with complex microstructures across scales from nanometers to meters. In this lecture a new data-driven computational framework for the analysis and design of these complex material systems will be presented. A mechanistic concurrent multiscale method called self-consistent clustering analysis (SCA) is developed for general inelastic heterogeneous material systems. The efficiency of SCA is achieved via data compression algorithms which group local microstructures into clusters during the off-line training stage, thereby reducing required computational expense. Its accuracy is guaranteed by introducing a self-consistent method for solving the Lippmann-Schwinger integral equation in the on-line predicting stage. The integration of microstructure reconstruction and subsequent high-fidelity multiscale predictions of the materials behavior leads to the generation of vast amounts of reliable data. This structure-property feedback loop enables the design of new material systems with new capabilities. In mathematical physics, the "structure" and "property" can be interpreted as the nonlocal interaction of the microstructure clusters and the virtual work at the corresponding material point, respectively. Based on the computational design of experiments, data mining techniques offer the ability to discover the influence of the microstructure on the macroscopic materials behavior. The proposed framework will be illustrated for advanced composites and the integrated design of various advanced material systems.

## Ricardo Nochetto: Two-scale FEMs for non-variational elliptic PDEs and pointwise convergence rates

We show that the finite element method (FEM) is able to approximate non-variational elliptic PDEs provided we add a larger scale ε to the usual meshsize h. We use the ε-scale to compute centered second differences of continuous functions which are piecewise linear at the h-scale, thereby replacing the notion of wide stencil while enforcing monotonicity without stringent mesh restrictions. We apply this basic principle to linear PDEs in non-divergence form, the Monge-Ampere equation, and a fully nonlinear PDE for the convex envelope. The scale ε plays a role similar to a finite horizon for integro-differential operators.

We derive pointwise error estimates for all three cases exploiting the separation of scales. A fundamental tool is a novel discrete Alexandroff estimate for continuous piecewise linear functions which states that the max-norm of their negative part is controlled by the Lebesgue measure of the sub-differential of their convex envelope at the contact nodes. We also develop a discrete Alexandroff-Bakelman-Pucci estimate which controls the Lebesgue measure of the sub-differential in terms of the discrete Laplacian via gradient jumps.

This is joint work with Wenbo Li, Dimitris Ntogkas and Wujun Zhang.

## Michael Ortiz: Deformation microstructure, relaxation and numerical approximation

We present a rigorous formulation of concurrent multiscale computing based on relaxation. We establish the connection between concurrent multiscale computing and enhanced-strain elements. We illustrate the approach in several area of applications, including martensite, single-crystal metal plasticity and energetic materials for which the explicit relaxation of the problem is derived analytically or computed on the fly. These example demonstrate the vast effect of microstructure formation on the macroscopic behavior of the material. Thus, whereas the unrelaxed model results in an overly stiff response, the relaxed model exhibits a proper limit load, as expected. Our numerical examples additionally illustrate that ad-hoc element enhancements, e.g., based on polynomial, trigonometric or similar representations, are unlikely to result in any significant relaxation in general.

## Mira Schedensack: Relaxing the CFL condition for the wave equation on adaptive meshes

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This talk shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

## Markus Schmuck: Porous media approximation of phase-field equations for interfacial dynamics

Consider the evolution of immiscible fluids [1,2,3] described by general homogeneous free energies in porous/strongly heterogeneous media. Using the homogenization method we systematically take the porosity and the pore geometry into account by deriving an effective upscaled phase field formulation. We expect that this new effective macroscopic formulation will have a broad range of applicability due to the generality of the underlying free energy formulation. We theoretically [4] and computationally [2,5] validate the effective porous media formulation by error estimates and we also recover the seemingly universal coarsening rate O(t^{1/3}) in the off-critical regime [5]. This universality seems to break down under thermal fluctuations (phase field equation + thermal noise) [5]. Hence, we expect that the analysis by homogenization allows for new, reliable modelling and computational perspectives in both, science and engineering.

References

[1] M. Schmuck, M. Pradas, G.A. Pavliotis, and S. Kalliadasis, Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains, Proc. R. Soc. A 468:3705-3724 (2012).

[2] M. Schmuck, M. Pradas, G.A. Pavliotis, and S. Kalliadasis, Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media, Nonlinearity 26(12):3259-3277 (2013).

[3] M. Schmuck, G.A. Pavliotis, and S. Kalliadasis, Effective macroscopic interfacial transport equations in strongly heterogeneous environments for general homogeneous free energies, Applied Mathematics Letters 35:12-17 (2014).

[4] M. Schmuck and S. Kalliadasis, Rate of Convergence of General Phase Field Equations in Strongly Heterogeneous Media towards their Homogenized Limit, submitted (2016).

[5] A. Veveris and M. Schmuck, Computational investigation of porous media phase field formulations: microscopic, effective macroscopic, and Langenvin equations, submitted (2016).

## Pablo Seleson: Concurrent Multiscale Methods for Local/Nonlocal Coupling

Nonlocal models have been proposed in recent years to overcome limitations of classical PDE-based (local) models. For instance, peridynamic models have been developed to simulate material failure and damage, whereas nonlocal (and fractional) diffusion models have beed advanced to properly describe anomalous diffusion and transport phenomena. These types of problems are challenging for classical local models. However, nonlocal models are computationally more expensive than their classical counterparts. Consequently, efforts have been made to develop concurrent multiscale methods to couple local and nonlocal models. Such methods provide means to simulate nonlocal problems with proper accuracy, but at a much cheaper computational cost. In mechanics, for instance, these methods allow one to employ peridynamic models in regions where cracks exist or may form, while using classical continuum mechanics models elsewhere, in regions where the deformation is smooth. In this presentation, we will provide an overview of local/nonlocal coupling in continuum mechanics, describe main challenges, and present coupling methods.

## Benjamin Stamm: A perturbation-method-based post-processing of planewave approximations for Density Functional Theory (DFT) models

In this talk, we consider a post-processing of planewave approximations for nonlinear Schrödinger equations by considering the exact solution as a perturbation of the discrete, computable solution. Applying then Kato’s perturbation theory leads to computable corrections with a provable increase of the convergence rate in the asymptotic range for a very little computational overhead for each considered eigenfunction and eigenvalue. We illustrate the mathematical key-features of this post-processing first for the Gross-Pitaevskii equation and then for DFT Kohn-Sham models. Finally numerical illustrations are presented.

## Xiaochuan Tian: Asymptotically Compatible Discretization of Nonlocal Variational Problems

Nonlocal continuum models are in general integral-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In this talk, we study robust finite element approximations of linear nonlocal diffusion and nonlocal mechanics models featured with a horizon parameter which characterizes the nonlocal interaction length. In particular, we give an abstract mathematical framework that can be used to identify asymptotically compatible finite element discretization schemes for nonlocal variational problems which give convergent schemes insensitive to parameter changing.

## Jeremy Trageser: Anisotropic Linearized Peridynamic Models

Peridynamics provides a method of modeling damage nucleation and propagation within various media; however, the majority of models focus on isotropic materials. This presentation will explore a generalized formulation of linearized peridynamic models capable of offering sufficient degrees of freedom to accommodate any material symmetry described by linear elasticity. We will further discuss the limitations bond-based peridynamic theory places on material symmetries and how the state-based theory removes these restrictions.

## Panayot Vassilevski: Spectral upscaling for graph Laplacian problems with application to reservoir simulation

To design coarse models that are both accurate and also easier to solve and require less memory than a better resolved (fine-scale) problem is important since it can help to reduce the cost of preforming many repeated simulations which may not be even feasible at the fine-scale. One way to achieve this goal is to employ specialized algebraic multigrid techniques viewed more as discretization rather than solution techniques. The latter is referred to as numerical upscaling. In this presentation, we use an abstract linear algebra setting, formulating the finite volume problem arising in oil reservoir simulation, as a graph Laplacian one. More specifically, we present coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We employ aggregation-based coarsening procedures applied to both sets of dofs to allow for more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. This allows for better local resolution (aggregate-by-aggregate). Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is an extension of the spectral upscaling procedure developed previously for the mixed finite element discretization of diffusion type PDEs which has the important property of maintaining inf-sup stability on coarse levels and having provable approximation properties. We consider applications to partitioning of a general graph and to a finite volume discretization interpreted as a graph Laplacian, demonstrating that the developed coarse-scale models possess the required properties for upscaling (accuracy, less memory) and hence can be used as accurate approximations to the fine-scale ones.

## Jiang Yang: Asymptotically compatible discretization of nonlocal models: quadrature-based finite difference and Fourier spectral methods

Many problems in nature, being characterized by a parameter, are of interests both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice. The asymptotically compatible schemes discussed here meet such objectives for a class of parametrized problems. In this talk, we will present two asymptotically compatible discretizations for nonlocal models, including specially designed quadrature based finite difference methods and Fourier spectral methods. In particular, we will discuss the efficient implementation of these two asymptotically compatible for nonlocal models. In the end, we will apply these asymptotically compatible discretizations for a robust a posteriori stress analysis, as well as some nonlocal gradient flows including nonlocal Allen-Cahn equations, nonlocal Cahn-Hilliard equations and nonlocal phase-field models.

## Wei Zhao: Localized radial basis functions-based pseudo-spectral method (LRBF-PS) for nonlocal diffusion problems

This is joint work with Yiuchung Hon and Martin Stoll. Spectral/pseudo-spectral methods based on high order polynomials have been successfully developed for solving partial differential and integral equations. In this talk, we will present the use of local radial basis functions-based pseudo-spectral method (LRBF-PS) for solving 2D nonlocal problems. The basic idea of LRBF-PS is to construct a set of orthogonal functions by RBF on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. Numerical implementation indicates that this proposal LRBF-PS method is simple to use, efficient and robust to solve various nonlocal problems. One of the main motivations for nonlocal problems is their ability to describe problems with discontinuities. These kinds of nonlocal problems can be tackled by combining the LRBF-PS with Hybrid scheme to avoid Gibbs phenomenon. Finally, several examples are given to verify the reliability and effectiveness of the proposal approaches.