Linear Nonlocal Problems Research Articles

Discretization schemes for the two simplified global double porosity models of immiscible incompressible two-phase flow

We present the discretization schemes for the two simplified homogenized models of immiscible incompressible two-phase flow in double porosity media with thin fractures. The two models were derived previously by the authors by different linearizations of the nonlinear local problem called the imbibition equation which appears in the homogenized model after passage to the limit as ε → 0. The models are fully homogenized with the matrix-fracture source terms expressed as a convolution.

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Breker layers in a pneumatic tire are an important part in the tire construction. These layers have a metal cord resulting in substantial bending stiffness. When homogenizing such layers, a “shave” method is applied to the breaker layer. This results in a thinner layer having adequate stiffness in both tension and bending. In this work, a phenomenological approach is used to obtain the effective properties of a homogeneous anisotropic hyper elastic material of an equivalent layer. Two models utilize transverse isotropic or orthotropic potential used to describe the homogenized properties. Comparison is made between these models for the “shaved” rubber-cord layer based on numerical experiments. In both cases, the potentials are built on the basis of the Treloar or Mooney potentials. Note that in the case of an inhomogeneous thin layer, the traditional definition of homogenization needs to be modified. In previous works of the authors, it was proposed to determine 3D averaged elastic properties of a layer by surrounding it with a homogeneous material. This makes it possible to correctly take into account the fact that the boundary effect from the upper to lower surfaces that penetrates through the whole periodicity cell. A set of local problems formulated for the periodicity cell is proposed. This set is sufficient for determining elastic potential material parameters. Nonlinear local problems on a periodic cell are solved and the material constants of the elastic potential are determined. The applicability of the orthotropic potential (second model) is determined for the “shaved” layer. It was found that orthotropic properties are manifested relative to longitudinal shears. The results show the suitability of the proposed potential and the scheme for determining the material parameters.

A novel second-order reduced homogenization approach for nonlinear thermo-mechanical problems of axisymmetric structures with periodic micro-configurations

A novel second-order reduced homogenization (SORH) approach is introduced for analyzing dynamic thermo-mechanical coupling problems in axisymmetric inelastic structures with periodic micro-configuration. The axisymmetric heterogeneous structures are periodically distributed in radial and axial directions and homogeneous distribution in circumferential directions. Firstly, the nonlinear coupled thermo-mechanical model is proposed, and the high-order nonlinear local problems, effective material parameters and the nonlinear homogenization equations are derived successively by the multiscale asymptotic expansion. Further, in order to reduce the large computational amount evaluated by the classical multiscale homogenization approach, the reduced-order nonlinear multiscale models and the corresponding finite-element algorithms are established in detail. The key features of the proposed approach are that an efficient reduced-model form based on transformation field analysis (TFA) to analyze nonlinear local cell problems is proposed and a nonlinear thermo-mechanical problem which considers the mutual coupling for the temperature and displacement fields is computed. In particular, a new SORH algorithm is proposed for investigating the axisymmetric inelastic structures. Finally, three typical numerical experiments are carried out, and the effectiveness and correctness of our presented algorithms in simulating and predicting the macroscopic behavior of the heterogeneous structures are confirmed.

Distributed networked localization using neighboring distances only through a computational topology control approach

For large-scale wireless sensor networks, the nonlinear localization problem where only neighboring distances are available to each individual sensor nodes have been attracting great research attention. In general, distributed algorithms for this problem are likely to suffer from the failures that localizations are trapped in local minima. Focusing on this issue, this article considers a fully distributed algorithm by introducing a novel mechanism, where each individual node is allowed to computationally interact with a random subset of its neighbors, for helping localizations escape from local minima. Theoretical analyses reveal that with the proposed algorithm, any local minimum of the localization will be unstable, and the global optimum would finally be achieved with probability 1 after enough time of iterations. Numerical simulations are given as well to demonstrate the effectiveness of the algorithm.

A global–local strategy with the generalized finite element framework for continuum damage models

This paper presents a global–local strategy with the generalized finite element framework to simulate structural failure through nonlinear continuum damage models. The global problem is the scale of the structure which is discretized with relative coarse meshes while the local problem is a subdomain of interest of the global problem where refined meshes can be used to simulate localized crack growth. The proposed method uses the converged nonlinear local problem solution as enrichment functions for the global problem which is considered linear with no need for an iterative procedure to solve it. The strategy is validated and compared with reference solutions and experimental results with crack propagation in mode I and mixed-mode conditions in monotonic tests of quasi-brittle materials. The results show that the method has the capability to transfer the kinematic effects and the damage state variable that occurs in the local problem to accurately predict the global structural behavior under the damage process. The proposed strategy demonstrated the capability to accurately predict the experimental crack paths and load–displacement curves with a reduced number of iterations and degrees of freedom in relation to the conventional finite element and generalized finite element methods.

SPACE-TIME NONLINEAR UPSCALING FRAMEWORK USING NONLOCAL MULTICONTINUUM APPROACH

In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients. The framework is built on a nonlinear nonlocal multicontinuum upscaling concept and significantly extends the results of earlier work. Our approach starts with a coarse space-time partition and identifies test functions for each partition, which play the role of multicontinua. The test functions are defined via optimization and play a crucial role in nonlinear upscaling. In the second stage, we solve nonlinear local problems in oversampled regions with some constraints defined via test functions. These local solutions define a nonlinear map from macroscopic variables determined with the help of test functions to the fine-grid fields. This map can be thought as a downscaled map from macroscopic variables to the fine-grid solution. In the final stage, we seek macroscopic variables in the entire domain such that the downscaled field solves the global problem in a weak sense defined using the test functions. We present an analysis of our approach for an example nonlinear problem. Our unified framework plays an important role in designing various upscaled methods. Because local problems are directly related to the fine-grid problems, it simplifies the process of finding local solutions with appropriate constraints. Using machine learning (ML), we identify the complex map from macroscopic variablesto fine-grid solution. We present numerical results for several porous media applications, including two-phase flow and transport.

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken’s dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.

A Hybridizable Discontinuous Galerkin Method for the $p$-Laplacian

We propose the first hybridizable discontinuous Galerkin method for the $p$-Laplacian equation. When using polynomials of degree $k \geq 0$ for the approximation spaces of $u$, $\boldsymbol{\nabla} u$, and $|\boldsymbol{\nabla} u|^{p-2}\boldsymbol{\nabla} u$, the method exhibits optimal $k+1$ order of convergence for all variables in $L^1$- and $L^p$-norms in our numerical experiments. For $k \geq 1$, an elementwise computation allows us to obtain a new approximation $u_h^*$ that converges to $u$ with order $k+2$. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of $u$ on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of...

Linear Theory for Self-Localization: Convexity, Barycentric Coordinates, and Cayley–Menger Determinants

Localization, finding the coordinates of an object with respect to other objects with known coordinates—hereinafter, referred to as anchors, is a nonlinear problem, as it involves solving circle equations when relating distances to Cartesian coordinates, or, computing Cartesian coordinates from angles using the law of sines. This nonlinear problem has been a focus of significant attention over the past two centuries and the progress follows closely with the advances in instrumentation as well as applied mathematics, geometry, statistics, and signal processing. The Internet-of-Things (IoT), with massive deployment of wireless tagged things, has renewed the interest and activity in finding novel, expert, and accurate indoor self-localization methods, where a particular emphasis is on distributed approaches. This paper is dedicated to reviewing a notable alternative to the nonlinear localization problem, i.e., a linear-convex method, based on Khan et al. ’s work. This linear solution utilizes relatively unknown geometric concepts in the context of localization problems, i.e., the barycentric coordinates and the Cayley–Menger determinants. Specifically, in an $m$ -dimensional Euclidean space, a set of $m+1$ anchors, objects with known locations, is sufficient (and necessary) to localize an arbitrary collection of objects with unknown locations—hereinafter, referred to as sensors, with a linear-iterative algorithm. To ease the presentation, we discuss the solution under a structural convexity condition, namely, the sensors lie inside the convex hull of at least $m+1$ anchors. Although rigorous results are included, several remarks and discussion throughout this paper provide the intuition behind the solution and are primarily aimed toward researchers and practitioners interested in learning about this challenging field of research. Additional figures and demos have been added as auxiliary material to support this aim.

Multidimensional scaling approach for node localization using received signal strength measurements

Node localization has played an important role in wireless sensor networks. In this paper, cooperative localization using received signal strength (RSS) measurements is addressed. The technique of weighted multidimensional scaling (WMDS) which relies on pairwise distance information between nodes is utilized in our algorithm development. Assuming that the transmit power is available, we first convert the original nonlinear localization problem to a system of linear equations, leading to computational attractiveness. It is also proved that the positioning accuracy of the WMDS solution attains the Cramér–Rao lower bound at sufficiently small noise conditions. Furthermore, the proposed method is extended to the unknown transmit power case by exploiting the ratio of squared distance estimates extracted from the RSS information. The effectiveness of the WMDS approach is demonstrated via comparison with several conventional RSS-based positioning methods.

A Framework for Low Complexity Least-Squares Localization With High Accuracy

In this paper, a new framework is proposed for least-squares localization based on estimated ranges, covering time-difference-of-arrival (TDoA), time-of-arrival (ToA), and received signal strength (RSS) cases. The multidimensional nonlinear localization problem is first transformed to a lower dimension and then solved iteratively. Within the proposed transformed least-squares (TLS) framework, we introduce a method in which the localization problem is transformed to one dimension (1-D). In this way, compared to the classical nonlinear least-squares (NLS) type of methods, the amount of computations in each iteration is greatly reduced; a reduction of 67% for a 3-D positioning system is shown. Hence, the introduced 1-D iterative (1DI) method is fairly light on the computational load. The way to choose the 1-D parameter is proposed, and theoretical expressions for the convergence rate and the root- mean-squared error (RMSE) of the 1DI estimator are derived. Validation is performed mainly based on actual ultra-wideband (UWB) radio measurements, collected in typical office environments, with signal bandwidths varying from 0.5 to 7.5 GHz. Supplementary simulations are also included for validation. Results show that, in terms of RMSE, the 1DI method performs better than the linear least-squares (LLS) method, where the solution is obtained noniteratively, and performs similarly as NLS, especially in TDoA cases.

Localization of unknown sources for parabolic systems on the basis of available observations

The paper considers two problems. Firstly the nonlinear localization problem; on the basis of available observations on the diffusion process, recover the location of an unknown single source generating this process. In general, the solution of this problem is set-valued and disconnected. Secondly, the identifiability problem; what types of observations (finite-dimensional at every moment of time, as generally occurs in physical situations) are able to provide enough information to restore the location point? A new approach is given, based on the introduction of a space of test-functions: in order to determine the unknown location, one needs to analyse a relevant system of algebraic equations. The latter can be determined in advance (‘off-line’). Sufficient conditions of identifiability are derived and the duality relations between the above nonlinear localization problems and the associated adjoint linear control problems are established. The above problems are motivated by environmental monitoring.