In the third part analysis and calculation according to the method of weighted residuals is presented, from establishing control equations and boundary conditions, selecting trial function, to calculating the stress and deflection loaded by wind-pressure or complex forces.
Beams and plates on Winkler foundation and elastic half space were calculated by collocation method and comparisons with theoretical results were made,which had indicated that this trial function was suited to solving many boundary value problems and also had a high accuracy.
From the governing equations of the elastic plane problem and the Airy stress function, the fundamental analytical solutions are first derived and then used as the trial functions to formulate the elements: ATF-Q4a, ATF-Q4b and ATF-Q4q.
with respect to the(N+1) points and two extended additional points in the x direction and the(M+1) points and two extended additional points in the y direction,are employed as the displacement trial functions.
From the governing equations of the elastic plane problem and the Airy stress function,the complete basic analytical solutions are derived and used as the trial functions to formulate the element: ATF-GCQ4X.
By starting from the general case of annular plates bearing the annular load, using the subdomain method in the weighted residual method,choosing simple trial functions (first, second, third-degree polynomial), we have found the solution of the limit load of annular plates on their various supporting conditions.
The meshless natural element method is a new numerical method based on the Voronoi diagram and dual Delaunay triangularization structure for a set of randomly distributed nodes. In the meshless natural element method, a local Petrov-Galerkin formulation is employed to abtain the discrete system equations,where trial functions are constructed using natural neighbour interpolants, and test functions are constructed using the finite element shape functions for a triangle with three nodes or its Bernstein-Bezier basis functions.
First, test functions selected to satisfy natural boundary conditions, but not to assure meeting equations, are regarded as the approximate solutions of equations and let weighted-integrating of errors in whole space domain equal to zero. It is explained that the sum of virtual work done by generalized forces equal to zero (the weak integration of equilibrium eqation).
In the study of exact elements,it has been shown and proved that,as long as the test functions are constructed using the solution of the adjoint differential equation,the element is bound to produce exact nodal solutions no matter what the trial functions are employed.
An effective procedure for solving the problem of plastic limit analysis of taper-shell with the method of weighted residuals is presented. By this procedure multinomials are used as a test function and subdomain method is applied to eliminate residuals.
This paper presents the strase analysis of rotating plates with trregular shapes. Thecontributions of displacements and stresses in the plates are obtained by the use of themathematical programming on method of weighted residual and the complete test function of double trigonometric series. The numerous results show that the stress contributions in rotating structuresof irregular shape are much different from regular structures.
The natural element method(NEM) is a newly appeared numerical method to solve the partial differential equation(PDE). In the natural element method,the trial and test function are constructed using the natural neighbor interpolation method and the interpolation is constructed with respect to the Voronoi tessellation of the scattered points in the problem domain.
In natural element method(NEM),the trial and test function are constructed with the natural neighbor interpolation(NNT) method. The interpolation is constructed with respect to the Voronoi tessellation of the scattered nodes in the problem domain.
The proof is based on the trial function method developed by Mitidieri and Pokhozhaev without recourse to comparison theorems and to the maximum principle.
When producing the trial function for the body trajectories in the "velocity-altitude" variables, we did not allow for fragmentation explicitly, since it is less probable for small meteoroids than for large ones.
Moreover, an analysis of the correlation between EL and EU with increasing number of terms in the expansion of the trial function makes it possible to improve the accuracy (at least by one order of magnitude) of the value E∞ extrapolated to infinity.
Making a Jastrow ansatz for a variational wave function of a strongly interacting extended Fermi system, we derive a set of integro-differential equations to relate the Jastrow trial function to the radial distribution function.
Our previously developed trial function scheme has been generalized to investigate the spin and charge polarization around the nearest neighbors of the impurity, and the influence of the polarization on the formation of local moments.
We obtain conditions for the nonexistence of global solutions and estimates of existence time for local solutions to the problem The proofs are based on the method of trial functions developed by Mitidieri and Pokhozhaev.
The variational principle for the integration of the convective-diffusion equation is developed, which reduces the problem solution to selecting adequate trial functions.
The application of the least-squares method with the use of various models as trial functions shows that the model of a single body with ablation provides the best fit to the observed trajectory.
Lower bounds with exponential trial functions are obtained for the first time (the corresponding formulas are presented for the first time as well); for a Gaussian basis, lower bounds for Coulomb systems have not been known either.
For variational calculations of molecular and nuclear systems involving a few particles, it is proposed to use carcass basis functions that generalize exponential and Gaussian trial functions.
In this article the p-adic Lizorkin spaces of test functions and distributions are introduced.
Methods of structural and structural-parametric design of complex nonlinear systems were developed on the basis of the method of test functions using the robust statistics and the Fokker-Plank-Kolmogorov equation.
The Boltzmann collision operator is computed for a variety of test functions characteristic of the motion of a rarefied gas and the values obtained according to it are compared to the Krook model.
Homological dimensions of certain algebras of principal (test) functions
The braking trajectories based on the model of successive destruction with ablation are used as the test functions.