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2019 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Peridynamic Differential Operator for Numerical Analysis

verfasst von: Dr. Erdogan Madenci, Atila Barut, Mehmet Dorduncu

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book introduces the peridynamic (PD) differential operator, which enables the nonlocal form of local differentiation. PD is a bridge between differentiation and integration. It provides the computational solution of complex field equations and evaluation of derivatives of smooth or scattered data in the presence of discontinuities. PD also serves as a natural filter to smooth noisy data and to recover missing data.

This book starts with an overview of the PD concept, the derivation of the PD differential operator, its numerical implementation for the spatial and temporal derivatives, and the description of sources of error. The applications concern interpolation, regression, and smoothing of data, solutions to nonlinear ordinary differential equations, single- and multi-field partial differential equations and integro-differential equations. It describes the derivation of the weak form of PD Poisson’s and Navier’s equations for direct imposition of essential and natural boundary conditions. It also presents an alternative approach for the PD differential operator based on the least squares minimization.

Peridynamic Differential Operator for Numerical Analysis is suitable for both advanced-level student and researchers, demonstrating how to construct solutions to all of the applications. Provided as supplementary material, solution algorithms for a set of selected applications are available for more details in the numerical implementation.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Although the differentiation process is usually more direct than integration in analytical mathematics, the reverse is true in computational mathematics, especially in the presence of a jump discontinuity or a singularity. Integration is a nonlocal process because it depends on the entire range of integration. However, differentiation is a local process. Mathematical modeling and understanding of most physical phenomena require the determination of derivatives of the field variable or the discrete data with or without scatter.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 2. Peridynamic Differential Operator
Abstract
This chapter describes the peridynamic differential operator (PDDO) introduced by Madenci et al. (2016, 2017). The PDDO employs the concept of PD interactions, and it is based on the orthogonality property of the PD functions. It restores the nonlocal interactions at a point by considering its association with the other points within an arbitrary domain of interaction. The PD differentiation recovers the local differentiation as this interaction domain approaches zero. It converts the local form of differentiation to its nonlocal PD form. It is simply a bridge between differentiation and integration. Therefore, the PDDO enables numerical differentiation through integration.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 3. Numerical Implementation
Abstract
Recasting the local differentiation by employing the nonlocal PDDO requires spatial integration which is not always amenable to analytical methods. Therefore, the integration is performed by using a meshless quadrature technique due to its simplicity. The domain is divided into a finite number of cells, each with a specific entity. The discretization may have a uniform or nonuniform structure. Prior to discretizing the differential equation and boundary conditions/initial conditions, the family (interaction domain) of each collocation point is formed, and its degree of interaction (weight function) with the family members is specified. Associated with a particular point, the integration is performed by summing the entity of the points within each family. The size of the family and the weight function can be different for each point. The size of the family may be established based on the computational efficiency; however, it should capture the characteristics of the differential equation.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 4. Interpolation, Regression, and Smoothing
Abstract
Interpolation and regression of data and smoothing of noisy data play a significant role in many scientific disciplines. Interpolation is an estimation of an unknown variable at output points (locations) by employing the known values at surrounding input points. Regression is an estimation of a variable at both input and output points by employing the known values at the surrounding input locations. Smoothing is an estimation of a variable at only known input points by employing the known input values. Smoothing may be necessary if the input data is noisy. It is worth noting that interpolation is different than regression and smoothing; the estimation based on interpolation passes through all the known input values. In other words, there is an exact recovery of the known values of the input points. There exist several methods for such estimations.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 5. Ordinary Differential Equations
Abstract
An ordinary differential equation (ODE) involves the derivatives of an unknown function that is dependent on a single independent variable. Its solution is usually constructed subject to a set of constraints referred to as initial or boundary conditions. Therefore, the ODEs are classified as initial value problems (IVP) and boundary value problems (BVP). In the case of a nonlinear ODE, the solution to its corresponding discrete form as nonlinear system of algebraic equations is achieved by using the Newton-Raphson method. The initial guess for the solution may be set to zero. The solution procedure is repeated until the relative error becomes less than the desired tolerance which is specified as ε = 10−5 throughout this chapter.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 6. Partial Differential Equations
Abstract
In partial differential equations (PDEs), the unknown function is dependent on more than one independent variable. While many linear PDEs can be solved analytically, the nonlinear PDEs may not be amenable to analytical solutions. Also, the computational challenge may become rather demanding when solving nonlinear PDEs. Second-order linear PDEs can be classified as parabolic, hyperbolic, and elliptic which may describe a diffusion process, wave propagation, and steady-state (time independent) phenomena, respectively. This chapter presents either implicit or explicit solutions to the following linear parabolic, hyperbolic, and elliptic equations and nonlinear elliptic equations:
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 7. Coupled Field Equations
Abstract
Modeling of physical processes may lead to coupled field equations arising from the presence of deformation, fluid flow, heat transfer, moisture diffusion, oxidation, electrical potential, and vacancy diffusion. Therefore, this chapter describes the application of PDDO for the solution of the following linear and nonlinear coupled field equations:
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 8. Integrodifferential Equations
Abstract
In an integrodifferential equation (IDE), the unknown function also appears under the integral sign, and it can be expressed in the formwhere u(x) is the unknown function and the parameter p denotes the degree of the derivative. The known functions K(x, t) and f(x) are referred to as the kernel and the forcing function, respectively. The limits of integration m(x) and n(x) can vary or remain as constants. The determination of the unknown function u(x) is achieved by enforcing the necessary initial conditions. According to the limits of integration, they are classified as Fredholm and Volterra IDEs. With constant limits of integration, it is classified as the Fredholm IDE of the form
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 9. Weak Form of Peridynamics
Abstract
This chapter presents the weak form of the peridynamic (PD) governing field equations. They specifically concern the Poisson’s equation and Navier’s equation under in-plane loading conditions. Their weak forms derived based on the variational approach enable the direct imposition of nonlocal essential and natural boundary conditions. The numerical solution to these equations can be achieved by considering either a uniform or a nonuniform discretization.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Chapter 10. Peridynamic Least Square Minimization
Abstract
This chapter presents the PD least square minimization (LSM) to construct the analytical expressions in integral form for PD approximation of a field variable and its derivatives on the basis of TSE and the moving LSM of error. Similar to the PDDO, it is also based on the concept of PD interactions. Unlike the PDDO, it does not require the construction of PD functions at each point.
Erdogan Madenci, Atila Barut, Mehmet Dorduncu
Backmatter
Metadaten
Titel
Peridynamic Differential Operator for Numerical Analysis
verfasst von
Dr. Erdogan Madenci
Atila Barut
Mehmet Dorduncu
Copyright-Jahr
2019
Electronic ISBN
978-3-030-02647-9
Print ISBN
978-3-030-02646-2
DOI
https://doi.org/10.1007/978-3-030-02647-9

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