Wave Dispersion and Basic Concepts of Peridynamics Compared to Classical Nonlocal Damage Models

2016 ◽  
Vol 83 (11) ◽  
Author(s):  
Zdeněk P. Bažant ◽  
Wen Luo ◽  
Viet T. Chau ◽  
Miguel A. Bessa
Keyword(s):  
Convergence Rates ◽  
Finite Difference Methods ◽  
Salient Feature ◽  
Wave Dispersion ◽  
Delta Function ◽  
Two Dimensions ◽  
Atomic Scale ◽  
Limit Case ◽  
Damage Models ◽  
Nonlocal Models

The spectral approach is used to examine the wave dispersion in linearized bond-based and state-based peridynamics in one and two dimensions, and comparisons with the classical nonlocal models for damage are made. Similar to the classical nonlocal models, the peridynamic dispersion of elastic waves occurs for high frequencies. It is shown to be stronger in the state-based than in the bond-based version, with multiple wavelengths giving a vanishing phase velocity, one of them longer than the horizon. In the bond-based and state-based, the nonlocality of elastic and inelastic behaviors is coupled, i.e., the dispersion of elastic and inelastic waves cannot be independently controlled. In consequence, the difference between: (1) the nonlocality due to material characteristic length for softening damage, which ensures stability of softening damage and serves as the localization limiter, and (2) the nonlocality due to material heterogeneity cannot be distinguished. This coupling of both kinds of dispersion is unrealistic and similar to the original 1984 nonlocal model for damage which was in 1987 abandoned and improved to be nondispersive or mildly dispersive for elasticity but strongly dispersive for damage. With the same regular grid of nodes, the convergence rates for both the bond-based and state-based versions are found to be slower than for the finite difference methods. It is shown that there exists a limit case of peridynamics, with a micromodulus in the form of a Delta function spiking at the horizon. This limit case is equivalent to the unstabilized imbricate continuum and exhibits zero-energy periodic modes of instability. Finally, it is emphasized that the node-skipping force interactions, a salient feature of peridynamics, are physically unjustified (except on the atomic scale) because in reality the forces get transmitted to the second and farther neighboring particles (or nodes) through the displacements and rotations of the intermediate particles, rather than by some potential permeating particles as on the atomic scale.

scholarly journals An Elementary Approximation of Dwell Time Algorithm for Ultra-Precision Computer-Controlled Optical Surfacing

Micromachines ◽  
2021 ◽  
Vol 12 (5) ◽  
pp. 471
Author(s):  
Yajun Wang ◽  
Yunfei Zhang ◽  
Renke Kang ◽  
Fang Ji
Keyword(s):  
Dwell Time ◽  
Convergence Rates ◽  
Rotational Symmetry ◽  
Time Algorithm ◽  
Two Dimensions ◽  
Efficient Computation ◽  
Deconvolution Problem ◽  
Ultra Precision ◽  
Computer Controlled Optical Surfacing ◽  
Computer Controlled

The dwell time algorithm is one of the key technologies that determines the accuracy of a workpiece in the field of ultra-precision computer-controlled optical surfacing. Existing algorithms mainly consider meticulous mathematics theory and high convergence rates, making the computation process more uneven, and the flatness cannot be further improved. In this paper, a reasonable elementary approximation algorithm of dwell time is proposed on the basis of the theoretical requirement of a removal function in the subaperture polishing and single-peak rotational symmetry character of its practical distribution. Then, the algorithm is well discussed with theoretical analysis and numerical simulation in cases of one-dimension and two-dimensions. In contrast to conventional dwell time algorithms, this proposed algorithm transforms superposition and coupling features of the deconvolution problem into an elementary approximation issue of function value. Compared with the conventional methods, it has obvious advantages for improving calculation efficiency and flatness, and is of great significance for the efficient computation of large-aperture optical polishing. The flatness of φ150 mm and φ100 mm workpieces have achieved PVr150 = 0.028 λ and PVcr100 = 0.014 λ respectively.

External optimal control of fractional parabolic PDEs

2020 ◽  
Vol 26 ◽  
pp. 20 ◽  
Author(s):  
Harbir Antil ◽  
Deepanshu Verma ◽  
Mahamadi Warma
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Convergence Rates ◽  
Complete Analysis ◽  
Very Weak Solutions ◽  
Parabolic Pdes ◽  
Nonlocal Models ◽  
Parabolic Case ◽  
Potential Benefits ◽  
Control Source

In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

scholarly journals A phase-field approximation of the Steiner problem in dimension two

2019 ◽  
Vol 12 (2) ◽  
pp. 157-179 ◽  
Author(s):  
Antonin Chambolle ◽  
Luca Alberto Davide Ferrari ◽  
Benoit Merlet
Keyword(s):  
Phase Field ◽  
Transportation Problem ◽  
Numerical Approximation ◽  
Unit Length ◽  
Field Approximation ◽  
Two Dimensions ◽  
Steiner Problem ◽  
Limit Case ◽  
Fixed Parameter ◽  
Γ Convergence

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form {1+\beta\,\theta}, where θ denotes the amount of transported mass and {\beta>0} is a fixed parameter (notice that the limit case {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {\{\mathcal{F}_{\varepsilon}\}} as {\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

scholarly journals Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
José Luis Gracia ◽  
Martin Stynes
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Convergence Rates ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Polynomial Solution ◽  
Difference Schemes ◽  
Smooth Functions ◽  
Numerical Performance ◽  
The Relationship

AbstractFinite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)≠ 0.

Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions

2011 ◽  
Vol 7 (9) ◽  
pp. 713-718 ◽  
Author(s):  
Stefan Heinze ◽  
Kirsten von Bergmann ◽  
Matthias Menzel ◽  
Jens Brede ◽  
André Kubetzka ◽  
...  
Keyword(s):  
Two Dimensions ◽  
Atomic Scale

scholarly journals High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

2015 ◽  
Vol 17 (2) ◽  
pp. 317-336 ◽  
Author(s):  
Wei Wang ◽  
Chi-Wang Shu ◽  
H.C. Yee ◽  
Dmitry V. Kotov ◽  
Björn Sjögreen
Keyword(s):  
Finite Difference ◽  
Multiple Scales ◽  
Finite Difference Methods ◽  
Propagation Speed ◽  
Two Dimensions ◽  
High Order ◽  
Reactive Flows ◽  
Fractional Step Method ◽  
High Order Finite Difference ◽  
Sr Method

AbstractIn this paper, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

NUMERICAL INTERFACES IN FINITE DIFFERENCE METHODS FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

1993 ◽  
Vol 01 (02) ◽  
pp. 151-184 ◽  
Author(s):  
TAO LIN
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Discontinuous Coefficients ◽  
Interface Problems ◽  
Artificial Boundary ◽  
Numerical Examples ◽  
Difference Methods

In this paper, we discuss the interface problems arising in using finite difference methods to solve hyperbolic equations with discontinuous coefficients. The schemes developed here can be used to handle four important types of numerical interfaces due to: (1) the discontinuity of the coefficients of the PDE, (2) using artificial boundary, (3) using different finite difference formulae in different areas, and (4) using different grid sizes in different areas. Stability analysis for these schemes is carried out in terms of conventional l1, l2, and l∞ norms so that the convergence rates of these schemes are obtained. Several numerical examples are supplied to demonstrate properties of these schemes.

3-D interpretation of reflected arrival times by finite‐difference techniques

Geophysics ◽  
1994 ◽  
Vol 59 (5) ◽  
pp. 844-849 ◽  
Author(s):  
M. Ali Riahi ◽  
Christopher Juhlin
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Real Data ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Computationally Efficient ◽  
Reflected Waves ◽  
Arrival Times ◽  
First Arrival ◽  
Difference Methods

Finite‐difference methods have generally been used to solve dynamic wave propagation problems over the last 25 years (Alterman and Karal, 1968; Boore, 1972; Kelly et al., 1976; and Levander, 1988). Recently, finite‐difference methods have been applied to the eikonal equation to calculate the kinematic solution to the wave equation (Vidale, 1988 and 1990; Podvin and Lecomte, 1991; Van Trier and Symes, 1991; Qin et al., 1992). The calculation of the first‐arrival times using this method has proven to be considerably faster than using classical ray tracing, and problems such as shadow zones, multipathing, and barrier penetration are easily handled. Podvin and Lecomte (1991) and Matsuoka and Ezaka (1992) extended and expanded upon Vidale’s (1988) algorithm to calculate traveltimes for reflected waves in two dimensions. Based on finite‐difference calculations for first‐arrival times, Hole et al. (1992) devised a scheme for inverting synthetic and real data to estimate the depth to refractors in the crust in three dimensions. The method of Hole et al. (1992) for inversion is computationally efficient since it avoids the matrix inversion of many of the published schemes for refraction and reflection traveltime data (Gjøystdal and Ursin, 1981).

Minimum traveltime calculation in 3-D graph theory

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1895-1898 ◽  
Author(s):  
Ningya Cheng ◽  
Leigh House
Keyword(s):  
Graph Theory ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Velocity Model ◽  
Two Dimensions ◽  
Prestack Migration ◽  
Runge Kutta Method ◽  
Path Search ◽  
Crucial Part ◽  
Shortest Path Search

Traveltime calculation is a crucial part of seismic migration schemes, especially prestack migration. There are many different ways to compute traveltimes. These methods can be divided into three categories: (1) Ray tracing (Julian and Gubbins, 1977; Červený et al., 1977). These treat the problem as a initial value problem by shooting rays from the source to the receivers. Or they can also treat the problem as a two‐point boundary value problem. An initial raypath is bent using perturbation theory until Fermat’s principle is satisfied. Nichols (1994) also computed traveltimes with the amplitude information attached to it in two dimensions. (2) Finite‐difference methods (Reshel and Kosloff, 1986; Vidale, 1988; van Trier and Symes, 1991). These solve the eikonal equation directly by using different numerical schemes such as the Runge‐Kutta method, wavefront expansion, or upwind finite difference. (3) Graph theory (Moser, 1991; Fisher and Lees, 1993; Meng et al. 1994). This method recasts the traveltime problem into a shortest path search over a network, which is constructed from the velocity model. This method is guaranteed to find a stable minimum traveltime with any velocity model.

Sign in / Sign up

Export Citation Format

Share Document