Approximate Methods of Higher Analysis

Householder, A. S.; Kantorovich, L. V.; Крылов, В. И.

doi:10.2307/2003014

Cited by 263 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These assumed expressions are then usually substituted into a variational statement of the original 3D model (usually the Cauchy continuum model) to carry out the integration through the small dimensions to reduce the original 3D formulation into a one-dimensional (1D) formulation in terms of the beam axis for beams, and 2D formulation for plates/shells. This is essentially the application of the Kantorovich method [16] to composite structures. When this method is used to construct the Kirchhoff-Love model for composite laminates, it is also called CLT To derive CLT for a composite laminate, the axiomatic method first introduces the so-called Kirchhoff-Love assumptions to express the displacement field as…”

Section: Kirchhoff-love Model Derived Using the Axiomatic Methodsmentioning

confidence: 99%

“…In strength of materials, we have learned the following formulas for beams made of a single isotropic material: (16) with EA denoting the extension stiffness, GJ the torsion stiffness, and EI 2 and EI 3 the bending stiffness about x 2 and x 3 , respectively. It is also noted here for a single isotropic material, E is the Young's modulus, G is the shear modulus, A is the cross-sectional area, J is the torsion constant, and I 2 and I 3 are the area moments of inertia about x 2 and x 3 , respectively.…”

Section: Euler-bernoulli Beam Modelmentioning

confidence: 99%

See 1 more Smart Citation

A Review of Modeling of Composite Structures

2024

Materials

View full text Add to dashboard Cite

This paper provides a brief review on modeling of composite structures. Composite structures in this paper refer to any structure featuring anisotropy and heterogeneity, including but not limited to their traditional meaning of composite laminates made of unidirectional fiber-reinforced composites. Common methods used in modeling of composite structures, including the axiomatic method, the formal asymptotic method, and the variational asymptotic method, are illustrated in deriving the classical lamination theory for the composite laminated plates. Future research directions for modeling composite structures are also pointed out.

show abstract

Section: Kirchhoff-love Model Derived Using the Axiomatic Methodsmentioning

confidence: 99%

Section: Euler-bernoulli Beam Modelmentioning

confidence: 99%

A Review of Modeling of Composite Structures

2024

Materials

View full text Add to dashboard Cite

show abstract

“…In view of the zero-asymptotic of the coefficients in system (22), it can be solved by implementing the simple reduction method [47] implying a finite number of terms in expressions (12). This leads to the following finite system of equations:…”

Section: Solution Constructionmentioning

confidence: 99%

“…In view of the zero‐asymptotic of the coefficients in system (), it can be solved by implementing the simple reduction method [47] implying a finite number of terms in expressions (). This leads to the following finite system of equations:

\begin{equation}\sum_{k = 1}^N {\sum_{\mu = 1}^4 {M_{m,k}^{[\lambda ,\mu ]}C_k^{[\mu ]}} } = K_m^{[\lambda ]},\;\;\;\lambda \, = 1,2,3,4,\;\;m\, = 1,2,\ldots,N.\end{equation}

…”

Section: Solution Constructionmentioning

confidence: 99%

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

Postolaki

Tokovyy

2023

Z Angew Math Mech

View full text Add to dashboard Cite

An inverse problem is solved for identifying unknown force loadings on the inner surface of a finite‐length hollow cylinder using the variational method of homogeneous solutions. The problem is considered within the axisymmetric formulation, and the radial and axial displacements of the outer surface of the cylinder are used as the auxiliary data for solving the inverse problem. The accessible surfaces of the cylinder (the end‐faces and the outer surface) are assumed to be free of force loading. By making use of the variational method of homogeneous solutions, the problem is reduced to an infinite system of linear algebraic equations. The solution is verified numerically and its stability with respect to small errors in the input data is analyzed.

show abstract

A New Method for Solving an Eigenvalue Problem for a System of Three Coulomb Particles within the Hyperspherical Adiabatic Representation

Abrashkevich

Kaschiev

Vinitsky

2000

Journal of Computational Physics

View full text Add to dashboard Cite

The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the onedimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.

show abstract

Approximate Methods of Higher Analysis

Cited by 263 publications

References 0 publications

A Review of Modeling of Composite Structures

A Review of Modeling of Composite Structures

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

A New Method for Solving an Eigenvalue Problem for a System of Three Coulomb Particles within the Hyperspherical Adiabatic Representation

Contact Info

Product

Resources

About