Critical analysis of the Ritz method in the case of natural boundary conditions and solution of some structural dynamics problems

1986 ◽  
Vol 79 (2) ◽  
pp. 575-575
Author(s):  
Ricardo O. Grossi
Keyword(s):  
Boundary Conditions ◽  
Structural Dynamics ◽  
Critical Analysis ◽  
Ritz Method ◽  
Natural Boundary ◽  
Natural Boundary Conditions ◽  
Dynamics Problems

A Novel and Accurate Ritz Formulation for Free Vibration of Rectangular and Skew Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
S. A. Eftekhari ◽  
A. A. Jafari
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Rectangular Plates ◽  
Natural Boundary ◽  
Boundary Points ◽  
Natural Boundary Conditions ◽  
Free Edges

One of the major limitations of the conventional Ritz method is its difficulty in implementation to the differential equations with natural boundary conditions at the boundary points/lines. Plates involving free edges/corners and irregularly shaped plates are two historical and classical examples which show that their solutions cannot be accurately approximated by the conventional Ritz method. To solve this difficulty, a simple, novel, and accurate Ritz formulation is introduced in this paper. It is revealed that the proposed methodology can produce much better accuracy than the conventional Ritz method for rectangular plates involving free edges/corners and skew plates.

scholarly journals Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

2020 ◽  
Vol 8 (4) ◽  
pp. 994-1010
Author(s):  
Weizhi Xu
Keyword(s):  
Boundary Conditions ◽  
Spline Interpolation ◽  
Partition Of Unity ◽  
Scattered Data ◽  
Rectangular Domain ◽  
Natural Boundary ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Boundary Conditions ◽  
Cubic Polynomials

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

Meshless Integral Method for Analysis of Elastoplastic Geotechnical Materials

2010 ◽  
Author(s):  
Jianfeng Ma ◽  
Joshua David Summers ◽  
Paul F. Joseph
Keyword(s):  
Boundary Conditions ◽  
Function Approximation ◽  
Integral Method ◽  
Weak Form ◽  
Constitutive Law ◽  
Governing Equations ◽  
Natural Boundary ◽  
Pressure Sensitive ◽  
Support Domain ◽  
Natural Boundary Conditions

The meshless integral method based on regularized boundary equation [1][2] is extended to analyze elastoplastic geotechnical materials. In this formulation, the problem domain is clouded with a node set using automatic node generation. The sub-domain and the support domain related to each node are also generated automatically using algorithms developed for this purpose. The governing integral equation is obtained from the weak form of elastoplasticity over a local sub-domain and the moving least-squares approximation is employed for meshless function approximation. The geotechnical materials are described by pressure-sensitive multi-surface Drucker-Prager/Cap plasticity constitutive law with hardening. A generalized collocation method is used to impose the essential boundary conditions and natural boundary conditions are incorporated in the system governing equations. A comparison of the meshless results with the FEM results shows that the meshless integral method is accurate and robust enough to solve geotechnical materials.

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