Nonlocal Balance Laws Research Articles

On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models

We present a convergence result from nonlocal to local behavior for a system of nonlocal balance laws. The velocity field of the underlying conservation laws is diagonal. In contrast, the coupling to the remaining balance laws involves a nonlinear right-hand side that depends on the solution, nonlocal term, and other factors. The nonlocal operator integrates the density around a specific spatial point, which introduces nonlocality into the problem. Inspired by multi-lane traffic flow modeling and lane-changing, the nonlocal kernel is discontinuous and only looks downstream. In this paper, we prove the convergence of the system to the local entropy solutions when the nonlocal operator (chosen to be of an exponential type for simplicity) converges to a Dirac distribution. Numerical illustrations that support the main results are also presented.

Optimal Control of Nonlocal Continuity Equations: Numerical Solution

The paper addresses an optimal ensemble control problem for nonlocal continuity equations on the space of probability measures. We admit the general nonlinear cost functional, and an option to directly control the nonlocal terms of the driving vector field. For this problem, we design a descent method based on Pontryagin’s maximum principle (PMP). To this end, we derive a new form of PMP with a decoupled Hamiltonian system. Specifically, we extract the adjoint system of linear nonlocal balance laws on the space of signed measures and prove its well-posedness. As an implementation of the designed descent method, we propose an indirect deterministic numeric algorithm with backtracking. We prove the convergence of the algorithm and illustrate its modus operandi by treating a simple case involving a Kuramoto-type model of a population of interacting oscillators.

Modeling Multilane Traffic with Moving Obstacles by Nonlocal Balance Laws

Modeling Multilane Traffic with Moving Obstacles by Nonlocal Balance Laws

On nonlocal geometrically exact shell theory and modeling fracture in shell structures

In this work, we have developed a nonlocal geometrically-exact shell theory and its computational formulation, which can model fracture and crack growth in shell structures under finite deformations. We have derived the local form of the nonlocal balance laws for a nonlocal continuum, which are instrumental in developing the nonlocal stress-resultant based geometrically-exact shell theory. This approach is particularly efficient for modeling material damage and structural failure process with strong discontinuities. A meshfree Galerkin weak formulation is developed for the nonlocal geometrically-exact shell theory by using nonlocal differential operators. Several numerical examples, including finite deformation and fractures, are conducted to verify the effectiveness of the present method. The numerical results demonstrate that the proposed nonlocal shell theory is an accurate and robust method to model the failure of shell structures.

A peridynamic Reissner‐Mindlin shell theory

SummaryIn this work, we have developed a state‐based peridynamics theory for nonlinear Reissener‐Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state‐based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.

Analysis of a system of nonlocal balance laws with weighted work in progress

We consider a system of nonlocal balance laws, modeling a multi-commodity supply edge. The coupling of the different goods is realized via a “weighted work in progress” (WWIP), such that the velocity of every balance law also depends on the “load” of the other commodities. This WWIP represents nonlocal impact and due to it, the proposed model offers it to weight goods which are close to the “entrance” different from goods which leave the supply edge. It also allows the nonlocal term to have an impact on the solution only on a subset of the space domain, changing over time. Incorporating weights and source terms in the right-hand side, we demonstrate the existence of weak solutions for the resulting model and, under some restrictions on the weight and the inhomogeneity, we ensure the uniqueness of weak solutions. Furthermore, we consider source terms which depend pointwise on the solution and prove existence of a solution under suitable conditions.

Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping

In this article, we generalize some existence and uniqueness results in [22] for scalar nonlocal balance laws to multi-dimensional nonlocal balance laws. Also in the multi-dimensional case it is possible to get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws but which we prove to be redundant in the nonlocal case. We start with a well known existence theorem for linear balance laws and the explicit formula for a solution to obtain a fixed-point problem in the nonlocal term. Using Banach's fixed-point theorem we prove existence and uniqueness of a fixed-point. This existence and uniqueness result is used for constructing and proving a unique solution of the multi-dimensional nonlocal balance law on a sufficiently small time horizon. While in many publications the nonlocal term will be considered over the entire space domain, we present an existence and uniqueness result which allows rather general integration areae for the nonlocal term. Eventually, we obtain a unique weak solution on every finite time horizon. Existence and uniqueness are complemented by stability results and an exhaustive study on more regular weak solutions in case the data is more regular.

Existence, uniqueness and regularity results on nonlocal balance laws

We consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions. The description “nonlocal” refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time. Existence of a weak solution for initial data and right hand side data in L1∩L∞, in L∞ and in special cases in L1 is shown via the method of characteristics, resulting in a fixed-point problem in the nonlocal term. The uniqueness of a weak solution with relatively weak assumptions on the flux function and the nonlocal term is established, so that the uniqueness result does not require the well-known “Kružkov” entropy condition as it is typical for (local) balance laws and was up to now used in the available literature also for nonlocal balance laws.

Nonlocal multi-scale traffic flow models: analysis beyond vector spaces

Realistic models of traffic flow are nonlinear and involve nonlocal effects in balance laws. Flow characteristics of different types of vehicles, such as cars and trucks, need to be described differently. Two alternatives are used here, \(L^p\)-valued Lebesgue measurable density functions and signed Radon measures. The resulting solution spaces are metric spaces that do not have a linear structure, so the usual convenient methods of functional analysis are no longer applicable. Instead ideas from mutational analysis will be used, in particular the method of Euler compactness will be applied to establish the well-posedness of the nonlocal balance laws. This involves the concatenation of solutions of piecewise linear systems on successive time subintervals obtained by freezing the nonlinear nonlocal coefficients to their values at the start of each subinterval. Various compactness criteria lead to a convergent subsequence. Careful estimates of the linear systems are needed to implement this program.

Nonlocal systems of balance laws in several space dimensions with applications to laser technology

For a class of systems of nonlinear and nonlocal balance laws in several space dimensions, we prove the local in time existence of solutions and their continuous dependence on the initial datum. The choice of this class is motivated by a new model devoted to the description of a metal plate being cut by a laser beam. Using realistic parameters, solutions to this model obtained through numerical integrations meet qualitative properties of real cuts. Moreover, the class of equations considered comprises a model describing the dynamics of solid particles along a conveyor belt.

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.

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.

Nonlocal continuum mechanics based on distributions

This paper presents a derivation of the balance laws of nonlocal continuum mechanics, based on the concept of microelements (very small collections of atoms or molecules). Mechanical fields (mass density, stress tensor, internal energy density, …) are assumed to be distributions (generalized functions), over a microelement surface and volume, influenced by distant microelements. Collection of these distributions in the limit when the surface and volume elements approach zero, defines the nonlocal mechanical fields. By this process, singularities and discontinuities that may be present in microelements, are smoothed out. The nonlocal balance laws are then established, having no nonlocal residuals.