ON ATTACHED MASSES OF PANELS OSCILLATING IN INCOMPRESSIBLE MEDIUM

2021 ◽  
pp. 56-64
Author(s):  
Valery A. BUZHINSKIY
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Oscillation Mode ◽  
Boundary Elements ◽  
Boundary Integral ◽  
Solar Array ◽  
Attached Mass ◽  
Linear Superposition ◽  
Incompressible Medium

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

On attached masses of panels oscillating in incompressible medium

2021 ◽  
pp. 56-64
Author(s):  
Valery A. BUZHINSKIY
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Oscillation Mode ◽  
Boundary Elements ◽  
Boundary Integral ◽  
Solar Array ◽  
Attached Mass ◽  
Linear Superposition ◽  
Incompressible Medium

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

scholarly journals The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup

2019 ◽  
Vol 11 (2) ◽  
pp. 463-474
Author(s):  
R.V. Shevchuk ◽  
I.Ya. Savka ◽  
Z.M. Nytrebych
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Boundary Integral ◽  
Kolmogorov Equation ◽  
Linear Parabolic Equation ◽  
One Dimensional ◽  
Wentzell Boundary Condition ◽  
Backward Kolmogorov Equation

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

scholarly journals NUMERICAL MODELLING OF DYNAMIC RESPONSE OF A PARTIALLY SATURATED POROELASTIC HALF-SPACE IN CASE OF A LOAD ACTING INSIDE A CUBIC CAVITY

2020 ◽  
Vol 82 (4) ◽  
pp. 507-523
Author(s):  
A.N. Petrov ◽  
M.V. Grigoryev
Keyword(s):  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Boundary Integral ◽  
Poroelastic Medium ◽  
Partially Saturated ◽  
Pore Pressures ◽  
The Laplace Transform ◽  
Poroelastic Material

Computer modeling based on the boundary element method is performed for the problem of loading in terms of the Heaviside step function inside a cubic cavity located in a partially saturated poroelastic half-space. A poroelastic medium is represented by a heterogeneous material-based model consisting of an elastic matrix phase and two phases of fillers – liquid and gas filling the pore system. The material model corresponds to a three-component medium. The constitutive relations of poroelastic medium written in terms skeleton displacements and pore pressures of fillers are considered. The original initial-boundary value problem is reduced to a boundary value problem by using the formal application of the Laplace transform. The research technique is based on the direct approach boundary integral equations of 3D isotropic linear theory of poroelasticity. Boundary integral equations corresponding to the boundary value problem are solved by the boundary element method in combination with the collocation method. In this study 8-noded elements have been adopted to discretize the boundary of poroelastic half-space. It is assumed that the element is linear with respect to displacements and pore pressures, while only one central node is used to represent tractions and fluxes. Algorithms for eliminating singularities, decreasing the order and subdividing elements are employed to compute the integral coefficients of a discrete analogue of the boundary integral equation. Regular integrals are calculated using the Gauss quadrature formula. The solution in time is obtained by numerical inversion of the Laplace transform. The numerical inversion method relies on quadrature formulas for computing the convolution integral. The time dependences of unknown displacement functions and pore pressures at points on the surface of the half-space and the cavity are plotted. The corresponding graphs are given. The influence of the cavity depth and degree of saturation on dynamic responses is investigated. The solution obtained by using the model of a fully saturated poroelastic material is compared to that of partially saturated poroelastic material. It is noted that the model used for solving this problem leads to an underestimation of displacement and overestimation of pore pressure estimates.

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

Boundary-Layer Similarity Near the Edge of a Rotating Disk

1975 ◽  
Vol 42 (3) ◽  
pp. 584-590 ◽  
Author(s):  
R. J. Bodonyi ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Approximate Solution ◽  
Flow Field ◽  
Boundary Value Problem ◽  
Asymptotic Expansions ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Rotating Disk ◽  
Angular Speed ◽  
Rotating Fluid

Numerical solutions of the similarity equations governing the flow near the edge of a finite rotating disk are found to be possible only for −2.06626 ≤ α ≤ 1, where α is the ratio of the disk’s angular speed to that of the rigidly rotating fluid far from the disk. Furthermore, for α ≤ −1 the solutions of the boundary-value problem are not unique, and along one of the solution branches a singular structure of the flow field is approached as α → −1. Using the method of matched asymptotic expansions an approximate solution is found along the singular branch which explains some of the problems encountered in finding numerical solutions.

scholarly journals Hydrodynamic interactions among multiple circular cylinders in an inviscid flow

2012 ◽  
Vol 712 ◽  
pp. 505-530 ◽  
Author(s):  
R. Sun ◽  
C. O. Ng
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Cyclic Permutation ◽  
Neumann Boundary ◽  
Hydrodynamic Interactions ◽  
Circular Cylinders ◽  
Constructive Method ◽  
Nonlinear Phenomena ◽  
Inviscid Fluid

AbstractHydrodynamic interactions among multiple circular cylinders translating in an otherwise undisturbed inviscid fluid are theoretically investigated. A constructive method for solving a Neumann boundary-value problem in a domain outside $N$ circles (one kind of Hilbert boundary-value problem in the complex plane) is presented in the study to derive the velocity potential of the liquid. The method employs successive offset functions combined with a ‘generalized cyclic permutation’ in turn to satisfy the impenetrable boundary condition on each circle. The complex potential is therefore expressed as $N$ isolated singularities in power series form and used to get instantaneous added masses of $N$ submerged circular cylinders. Then, based on the Hamilton variational principle, a dynamical equation of motion in vector form is derived to predict nonlinear translations of the submerged bodies under fully hydrodynamic interactions. Also, the equivalence of the energy-based Lagrangian framework and a momentum-type one in the two-dimensional body–liquid system is proved. It implies that the pressure integration around a submerged body is holographic, which provides information about velocities and accelerations of all bodies. The numerical solutions indicate some typical dynamical behaviours of more than two circular cylinders which reveal that interesting nonlinear phenomena would appear in such a system with simple physical assumptions.

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