Nonlocal Calculus Research Articles

Nonlocal geometric fracture theory: A nonlocal‐local asymptotically compatible framework for continuous–discontinuous deformation of solid materials

SummaryOne of the fundamental but unsolved problems in fracture mechanics of solid materials is how to describe the continuous and discontinuous deformation of continuum in a unified manner. This paper aims to develop a novel nonlocal geometric fracture theory attempting to address this essential issue. Using the concepts of nonlocal calculus, a new nonlocal equation for the deformation of continuum is obtained by minimizing a nonlocal Lagrangian. This is done to ensure that the proposed nonlocal model can be used to describe the continuous and discontinuous deformation of a continuum in a unified geometric manner, as well as to be asymptotically compatible with the classical continuum theory. Inspired by the phase field fracture theory, a nonlocal geometric crack surface functional is proposed to approximate the Griffith fracture energy in continuum. Note, however, that a distinguished feature of the proposed functional is that it enables strong discontinuities appear in the crack phase fields, which is difficult for the phase field fracture theory. After obtaining the nonlocal fracture energy, a nonlocal elastic energy is phenomenologically decomposed into volumetric and deviatoric parts, and then a system of nonlocal geometric fracture equations is derived to describe the damage nucleation, cracks initiation and propagation, which covers the whole physical process of fracture in solid materials. A staggered discontinuous Galerkin method is developed to reach an accurate and stable numerical solution. Numerical examples demonstrate the proposed theory can well capture the transition from the continuous to discontinuous deformation of the continuum and accurately describe the topological evolution of cracks in a unified framework.

Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modeling of dynamical systems

We present an approach to reduced-order modeling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501–530, 2019). We extend a non-local calculus on finite weighted graphs to build such models by exploiting polynomial expansions and Taylor series. In the general framework for non-local calculus on graphs, the graph edge weights are intricately linked to the embedding of the graph, and consequently to the definition of the derivatives. In a previous communication (Duschenes and Garikipati, arXiv:2105.01740 [math.NA]), we have shown that radially symmetric, continuous edge weights derived from, for example Gaussian functions, yield inconsistent results in the resulting non-local derivatives when compared against the corresponding local, differential derivative definitions. Taking inspiration from finite difference methods, we algorithmically compute edge weights, considering the embedding of the local neighborhood of each graph vertex. Given this procedure, we ensure the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that we can achieve any desired orders of accuracy of derivatives, in a chosen number of dimensions without symmetry assumptions in the underlying data. Finally, we present two example applications of extracting reduced-order models using this non-local calculus, in the form of ordinary differential equations from parabolic partial differential equations of progressively greater complexity.

A new nonlocal calculus framework. Helmholtz decompositions, properties, and convergence for nonlocal operators in the limit of the vanishing horizon

A new nonlocal calculus framework. Helmholtz decompositions, properties, and convergence for nonlocal operators in the limit of the vanishing horizon

Nonlocal trace spaces and extension results for nonlocal calculus

For a given Lipschitz domain Ω, it is a classical result that the trace space of W1,p(Ω) is W1−1/p,p(∂Ω), namely any W1,p(Ω) function has a well-defined W1−1/p,p(∂Ω) trace on its codimension-1 boundary ∂Ω and any W1−1/p,p(∂Ω) function on ∂Ω can be extended to a W1,p(Ω) function. In this work, we study function spaces for nonlocal Dirichlet problems involving integrodifferential operators with a finite range of nonlocal interactions, and provide a characterization of their trace spaces. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical W1−1/p,p(∂Ω) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.

Optimization and learning with nonlocal calculus

<p style='text-indent:20px;'>Nonlocal models have recently had a major impact in nonlinear continuum mechanics and are used to describe physical systems/processes which cannot be accurately described by classical, calculus based "local" approaches. In part, this is due to their multiscale nature that enables aggregation of micro-level behavior to obtain a macro-level description of singular/irregular phenomena such as peridynamics, crack propagation, anomalous diffusion and transport phenomena. At the core of these models are <i>nonlocal</i> differential operators, including nonlocal analogs of the gradient/Hessian. This paper initiates the use of such nonlocal operators in the context of optimization and learning. We define and analyze the convergence properties of nonlocal analogs of (stochastic) gradient descent and Newton's method on Euclidean spaces. Our results indicate that as the nonlocal interactions become less noticeable, the optima corresponding to nonlocal optimization converge to the "usual" optima. At the same time, we argue that nonlocal learning is possible in situations where standard calculus fails. As a stylized numerical example of this, we consider the problem of non-differentiable parameter estimation on a non-smooth translation manifold and show that our <i>nonlocal</i> gradient descent recovers the unknown translation parameter from a non-differentiable objective function.</p>

Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

A non-smooth non-local variational approach to saliency detection in real time

In this paper, we propose and solve numerically a general non-smooth, non-local variational model to tackle the saliency detection problem in natural images. In order to overcome the typical drawback of the non-local methods in image processing, which mainly is the inherent computational complexity of non-local calculus, as the non-local derivatives are computed w.r.t every point of the domain, we propose a different scenario. We present a novel convex energy minimization problem in the feature space, which is efficiently solved by means of a non-local primal-dual method. Several implementations and discussions are presented taking care of the computing platforms, CPU and GPU, achieving up to 33 fps and 62 fps respectively for 300 $$\times$$ 400 image resolution, making the method eligible for real time applications.

Forced oscillation of fractional differential equations via conformable derivatives with damping term

Based on the properties of nonlocal fractional calculus generated by conformable derivatives, we establish some sufficient conditions for oscillation of all solutions for fractional differential equations with damping term. Forced oscillation of conformable differential equations in the frame of Riemann, as well as of Caputo type, is established. Examples are provided to demonstrate the effectiveness of the main results.

A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

We introduce here a nonlocal operator as a natural generalization to the biharmonic operator that appears in plate theory. This operator is built in the nonlocal calculus framework defined by Du et al. and its connected with the recent theory of peridynamics. For the steady state equation coupled with different boundary conditions we show existence and uniqueness of solutions, as well as regularity of solutions. The boundary conditions considered are nonlocal counterparts of the classical clamped and hinged boundary conditions. For each system we show convergence of the nonlocal solutions to their local equivalents using compactness arguments developed by Bourgain, Brezis and Mironescu.

A Class of High Order Nonlocal Operators

We study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators. We present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of Sobolev inequalities such as the nonlocal Poincare and nonlocal Gagliardo–Nirenberg inequalities. Nonlocal characterizations of high order Sobolev spaces in the spirit of Bourgain–Brezis–Mironescu are provided. Applications of nonlocal calculus of variations to the well-posedness of linear nonlocal models of elastic beams and plates are also considered.

From peridynamics to stochastic jump process: Illustrations of nonlocal balance laws and nonlocal calculus framework

Nonlocal effects are ubiquitous in nature. There is a need to go beyond the traditional setup to develop new mathematical insight and computational methods for their systematics study and understanding. In this paper, by working with peridynamic models of nonlocal mechanics and nonlocal diffusion models for stochastic jump processes, we systematically explore the mathematical description of nonlocal balance laws. We use simple examples to introduce the recently developed concepts of nonlocal vector calculus and nonlocal calculus of variations and to illustrate the connections to traditional local models and the key differences. We also present the notion of asymptotically compatible (AC) discretization schemes as robust numerical algorithms for simulating multiscale problems.

Formula omitted] schemes for finite element discretization of the space–time fractional diffusion equations

The numerical solution of a space–time fractional diffusion equation used to model the anomalous diffusion is considered. Spatial discretization is effected using a finite element method whereas the θ-scheme is used for temporal discretization. The fully discrete scheme is analyzed for all 0≤θ≤1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution. The analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus. One-dimensional numerical examples are provided that illustrate the theoretical stability and convergence results.

A generalized nonlocal vector calculus

A nonlocal vector calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A formulation is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green’s identities. The nonlocal vector calculus introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.

A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS

A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is the posing of abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics.

A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems

We develop a calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical vector calculus. The operators we define do not involve derivatives. We then apply the nonlocal calculus to define weak formulations of nonlocal “boundary-value” problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal problems, we derive a fundamental solution and Green's functions, demonstrate that weak formulations of the nonlocal “boundary-value” problems are well posed, and show how, under appropriate limits, the nonlocal problems reduce to their local analogues.

Optimal control of memory dependent nonlocal elastic solids

Sensitivity expressions of a general performance criterion with respect to load control functions have been derived for memory dependent nonlocal elastic solids. The memory dependence of the media has been taken in a hereditary integral form. By using nonlocal calculus of variations and the adjoint variable method of optimization, an adjoint problem is constructed in order to find the first variation of the performance criterion (i.e. a nonlocal and memory dependent integral functional) in terms of variations of the load control variables. An unconstrained optimal control example problem is provided whose solution has been obtained by using the sensitivity analysis results and also by a direct evaluation and differentiation of the relevant objective function, thereby showing the validity of the general sensitivity results derived in the present paper.

Fundamental necessary conditions of non-local calculus of variations

Abstract

Invariance theory for nonlocal variational principles—I extended variations

This is the first in a series of four papers dealing with the extensions of the Noether theorems on invariance and conservation laws to the nonlocal calculus of variations. After a summary of the germain aspects of the nonlocal variational calculus, the effects of the action of coordinate transformations and point transformations on the domain space of dependent functions are obtained. These results are then used to compute the variations in a functional of functionals that are induced by infinitesimal coordinate transformations and point transformations. Several forms are obtained for such variations since each will be required in the sequel.

Operation separable problems in the nonlocal calculus of variations

Operation separable problems in the nonlocal calculus of variations