The generation of non-ordinary state-based peridynamics by the weak form of the peridynamic method

2020 ◽  
Vol 25 (8) ◽  
pp. 1544-1567
Author(s):  
Hao Cui ◽  
Chunguang Li ◽  
Hong Zheng
Keyword(s):  
Differential Operators ◽  
Order Approximation ◽  
Weak Form ◽  
Higher Order ◽  
Zero Energy ◽  
Numerical Tests ◽  
Energy Mode ◽  
Stress Point ◽  
Bifurcation Problems ◽  
Special Case

A weak form of the peridynamic (PD) method derived from the classical Galerkin framework by substituting the traditional derivatives into the PD differential operators is proposed. The attractive features of the proposed weak form of PD method include the following: (1) a higher-order approximation than the non-ordinary state-based peridynamic (NOSB-PD) in the strain construction; (2) the NOSB-PD is demonstrated as a special case of the weak form of the PD method; (3) as an extension of the NOSB-PD, the zero-energy mode oscillations in the weak form of the PD can be significantly reduced by introducing higher-order PD derivatives. In addition, a series of numerical tests are conducted. The results show the following: (1) the three proposed stabilization items containing higher-order PD derivatives have a better accuracy and stability than the traditional items of the NOSB-PD. In particular, the stress point stabilization item is preferred since it has the highest accuracy and efficiency and does not introduce any additional parameters; (2) the weak form of PD method is very suitable in dealing with the crack propagation and bifurcation problems.

A Higher Order Approximation for the Electronic Structure of Hydrogen in Metals

1986 ◽  
Vol 41 (12) ◽  
pp. 1381-1398
Author(s):  
K. Göbel ◽  
F. Wahl
Keyword(s):  
Order Approximation ◽  
Energy Eigenvalue ◽  
Energy Difference ◽  
Electronic Energy ◽  
Host Lattice ◽  
Higher Order ◽  
Logarithmic Singularities ◽  
Special Case ◽  
Storage Energy ◽  
Coulomb Potentials

A functional energy difference method based on first principles is used to calculate the electronic contribution to the storage energy of hydrogen impurities in metals. That electronic problem is treated in a higher order approximation which means considering the coupling between one-electron wavefunctions and three-particle amplitudes. A formalism is presented to eliminate all higher order correlations and to reduce the whole system to a one-particle Schrödinger equation with the help of a suitable G reen’s function. The resulting one-electron eigenvalue problem contains some logarithmic singularities due to the fact that there is no gap between occupied and unoccupied electron states in the band structure of a metal. In an electron gas model a convergent theory is reached by an improvement of the Green’s function leading to a screening of the long-range Coulomb potentials. The result is a complicated non-linear algebraic eigenvalue equation which is solved numerically for the special case of a single hydrogen perturbation in a magnesium crystal. The solution shows the influence of higher order electron correlations on the electronic energy eigenvalue of an interstitial hydrogen centre plot as a function of host lattice distortions.

The Dual-Weighted Residual Estimator Realized on Polygonal Meshes

2018 ◽  
Vol 18 (4) ◽  
pp. 753-776 ◽  
Author(s):  
Steffen Weißer ◽  
Thomas Wick
Keyword(s):  
Order Approximation ◽  
Partition Of Unity ◽  
Weak Form ◽  
Single Element ◽  
Weighted Residual Method ◽  
Great Flexibility ◽  
Polygonal Meshes ◽  
Numerical Tests ◽  
Strong Localization ◽  
Discrete Problems

AbstractIn this work, we realize goal-oriented error estimation using the dual-weighted residual method on general polygonal meshes. Such meshes are of current interest in various applications thanks to their great flexibility. Specifically the discrete problems are treated on BEM-based FEM. Our dual-weighted residual estimator is derived for two localization procedures. Firstly, a classical (strong) localization. Secondly, a weak form is adopted in which localization is achieved with the help of a partition-of-unity. The dual (i.e., adjoint) solution is obtained via a local higher-order approximation using a single element. Our algorithmic developments are substantiated with the help of several numerical tests.

scholarly journals Self-updated four-node finite element using deep learning

2021 ◽  
Author(s):  
Jaeho Jung ◽  
Hyungmin Jun ◽  
Phill-Seung Lee
Keyword(s):  
Finite Element ◽  
Deep Learning ◽  
Iterative Solution ◽  
Solution Procedure ◽  
Element Formulation ◽  
Zero Energy ◽  
Energy Mode ◽  
Bending Modes ◽  
Solution Accuracy ◽  
Iterative Solution Procedure

AbstractThis paper introduces a new concept called self-updated finite element (SUFE). The finite element (FE) is activated through an iterative procedure to improve the solution accuracy without mesh refinement. A mode-based finite element formulation is devised for a four-node finite element and the assumed modal strain is employed for bending modes. A search procedure for optimal bending directions is implemented through deep learning for a given element deformation to minimize shear locking. The proposed element is called a self-updated four-node finite element, for which an iterative solution procedure is developed. The element passes the patch and zero-energy mode tests. As the number of iterations increases, the finite element solutions become more and more accurate, resulting in significantly accurate solutions with a few iterations. The SUFE concept is very effective, especially when the meshes are coarse and severely distorted. Its excellent performance is demonstrated through various numerical examples.

scholarly journals Lieb–Thirring inequalities for higher order differential operators

2008 ◽  
Vol 281 (2) ◽  
pp. 199-213 ◽  
Author(s):  
Clemens Förster ◽  
Jörgen Östensson
Keyword(s):  
Differential Operators ◽  
Higher Order

scholarly journals A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Weiping Shen
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Eigenvalue Problems ◽  
Generalized Newton Method ◽  
Inexact Newton Method ◽  
Numerical Tests ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
Generalized Newton ◽  
Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

Sign in / Sign up

Export Citation Format

Share Document