Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators

2009 ◽  
Vol 33 (4) ◽  
pp. 514-521 ◽  
Author(s):  
C.C. Tsai ◽  
A.H.-D. Cheng ◽  
C.S. Chen
Keyword(s):  
Particular Solutions ◽  
Helmholtz Operators

scholarly journals Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Chia-Cheng Tsai
Keyword(s):  
Present Method ◽  
Thick Plate ◽  
Mindlin Plate ◽  
Shear Forces ◽  
Helmholtz Operator ◽  
Particular Solutions ◽  
Product Operator ◽  
Plate Models ◽  
Operator Decomposition ◽  
Helmholtz Operators

Analytical particular solutions of the polyharmonic multiquadrics are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.

scholarly journals Bilinear discrete Painleve-II and its particular solutions

1995 ◽  
Vol 28 (12) ◽  
pp. 3541-3548 ◽  
Author(s):  
J Satsuma ◽  
K Kajiwara ◽  
B Grammaticos ◽  
J Hietarinta ◽  
A Ramani
Keyword(s):  
Particular Solutions ◽  
Painlevé Ii

General Features of Plastic-Elastic Problems as Exemplified by Some Particular Solutions

1949 ◽  
Vol 16 (3) ◽  
pp. 295-300
Author(s):  
Rodney Hill
Keyword(s):  
Spherical Shell ◽  
Infinite Medium ◽  
Cylindrical Hole ◽  
Particular Solutions ◽  
Complete Solutions ◽  
Complex Cases

Abstract New complete solutions based upon the Reuss equations are obtained for various plastic-elastic problems. These include the expansion of a spherical shell and of a cylindrical hole in an infinite medium. The solutions are used to exemplify certain features common to all plastic-elastic problems, with a view to introducing valid approximations in more complex cases.

Analytical Calculation of Deformations and Kinematic Analysis of a Flat Truss with an Arbitrary Number of Panels

2021 ◽  
pp. 113-122
Author(s):  
М. Н. Кирсанов ◽  
О. В. Воробьев
Keyword(s):  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Analytical Calculation ◽  
Middle Part ◽  
Asymptotic Approximations ◽  
Induction Method ◽  
Particular Solutions

Постановка задачи. Разыскиваются аналитические зависимости прогиба и смещения опоры плоской фермы решетчатого вида от числа панелей. Ферма имеет сдвоенную решетку, прямолинейный нижний и приподнятый в средней части верхний пояс. Результаты. Для двух видов нагружения по формуле Максвелла-Мора получены аналитические зависимости прогибов конструкции от нагрузки, размеров и числа панелей. Для обобщения серии частных решений с различным числом панелей ферм на произвольный случай использован метод индукции и аналитические возможности системы компьютерной математики Maple. Для некоторых решений получены асимптотические приближения. Показано распределение усилий в элементах фермы. Выводы. Полученные формулы могут быть использованы в задачах оптимизации и как тестовые для оценки приближенных численных решений. Выявлены случаи геометрической изменяемости фермы при числе панелей, кратном трем. Приведен алгоритм выявления соответствующего распределения возможных скоростей шарниров. Statement of the problem. Analytical dependences of the deflection and displacement of the support of a flat lattice truss on the number of panels are being sought. The truss has a double lattice, a rectilinear lower belt and an upper belt raised in the middle part. Results. For two types of loading, according to the Maxwell-Mohr formula, analytical dependences of the deflections of the structure on the load, dimensions and number of panels are obtained. To generalize a series of particular solutions for trusses with different numbers of panels for an arbitrary case, the induction method and the analytical capabilities of the Maple computer mathematics system were used. For some solutions, asymptotic approximations are obtained. The distribution of forces in the rods of the structure is shown. Conclusions. The obtained formulas can be used in optimization problems and as test ones for evaluating approximate numerical solutions. Cases of geometric variability of the truss with the number of panels being a multiple of three are revealed. An algorithm for identifying the corresponding distribution of possible velocities of the joints is presented.

Stability of Thin Elastic Plates Covering an Arbitrary Simply Connected Region and Subject to any Admissible Boundary Conditions

1953 ◽  
Vol 20 (1) ◽  
pp. 23-29
Author(s):  
G. A. Zizicas
Keyword(s):  
Boundary Conditions ◽  
Direct Method ◽  
Elastic Plates ◽  
Simple Manner ◽  
Particular Solutions ◽  
Isotropic Plates ◽  
Simply Connected Region ◽  
Simply Connected ◽  
A Direct Method ◽  
Thin Elastic Plates

Abstract The Bergman method of solving boundary-value problems by means of particular solutions of the differential equation, which are constructed without reference to the boundary conditions, is applied to the problem of stability of thin elastic plates of an arbitrary simply connected shape and subject to any admissible boundary conditions. A direct method is presented for the construction of particular solutions that is applicable to both anisotropic and isotropic plates. Previous results of M. Z. Krzywoblocki for isotropic plates are obtained in a simple manner.

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