A formulation for elastodynamic scattering based on boundary algebraic equations

2017 ◽  
Vol 141 (5) ◽  
pp. 4034-4034
Author(s):  
Jordi Poblet-Puig ◽  
Andrey V. Shanin
Keyword(s):  
Algebraic Equations ◽  
Boundary Algebraic Equations

scholarly journals Boundary algebraic equations for lattice problems

2009 ◽  
Vol 465 (2108) ◽  
pp. 2489-2503 ◽  
Author(s):  
Per-Gunnar Martinsson ◽  
Gregory J. Rodin
Keyword(s):  
Boundary Value Problems ◽  
Spectral Properties ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Boundary Integral ◽  
Linear Boundary ◽  
Algebraic Equations ◽  
Problem Size ◽  
Boundary Algebraic Equations ◽  
Fundamental Boundary

Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.

On Extension of Boundary Algebraic Equations to Irregular Lattices

2011 ◽  
Author(s):  
Haifeng Zhao ◽  
Gregory J. Rodin
Keyword(s):  
Boundary Integral Equations ◽  
Computational Effort ◽  
Boundary Integral ◽  
Future Research ◽  
Algebraic Equations ◽  
Infinite Domains ◽  
Multiscale Problems ◽  
Regular Lattices ◽  
Boundary Algebraic Equations ◽  
Interesting Direction

This study is concerned with extending the use of boundary algebraic equations (BAEs) to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as BAEs provide seamless bridging between discrete and continuum models. BAEs share many features with boundary integral equations and are particularly effective for solving problems involving infinite domains. However, it is shown that, BAEs for irregular lattices containing certain terms may require the same amount of computational effort as the original problem for which the equations are formulated. In this paper, we formulate a BAE for a model problem and expose the fundamental obstacle that prevents us from using that BAE as an effective tool. It is shown that, in contrast to regular lattices, BAEs for irregular lattices require a statistical rather than deterministic treatment. This is a very interesting direction for future research.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

MODELING OF NONLINEAR VIBRATION PROBLEMS WITH FLUID FLOWS THROUGH PIPELINES

2018 ◽  
pp. 44-47
Author(s):  
F.J. Тurayev
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Viscoelastic Properties ◽  
Construction Material ◽  
Fluid Flows ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Flowing Liquid ◽  
Vibration Problems

In this paper, mathematical model of nonlinear vibration problems with fluid flows through pipelines have been developed. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integro-differential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integro-differential equations with both constant and variable coefficients as functions of time.A system of algebraic equations is obtained according to numerical method for the unknowns. The influence of the singularity of heredity kernels on the vibrations of structures possessing viscoelastic properties is numerically investigated.It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

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