Interpretation Of Boundary Conditions Research Articles

Interpretation of boundary conditions in the analytical and numerical shell solutions for mode analysis of multilayered structures

The paper proposes the first 18 vibration modes for plates, and the first 14 vibration modes for cylinders and cylindrical shells. All the edges of these structures are simply supported and the free frequencies are calculated using an exact three-dimensional shell model. A comparison is proposed using two different numerical models such as a classical two-dimensional finite element model and a refined two-dimensional generalized differential quadrature model. The 3D exact model gives all types of vibration modes, when the four edges are simply supported, changing the imposed half-wave numbers m and n in the two in-plane directions α and β. Some of these modes have one of the two half-wave numbers equals zero. When this condition is simultaneously combined with the condition of transverse displacement different from zero, the resulting vibration mode is defined as cylindrical bending mode. The cylindrical bending case has all the derivatives made in the direction where m=0 or n=0 equal zero. This feature means that the vibrational behavior does not change along this particular direction. The numerical models with the simply supported boundary conditions for all the edges do not achieve these results. These cylindrical bending numerical results are obtained modifying the boundary conditions. Proposed results will demonstrate the validity of this idea and how to modify the mathematical models in order to obtain and improve the cylindrical bending solutions.

On proper formulation of boundary condition for degenerated PDEs when trace embedding theorems are missing and application to nonhomogeneous BVPs

We propose certain interpretation of boundary conditions on , the boundary of , when f belongs to the weighted Sobolev space subordinated to the integrable weight and g is defined on , where is a given domain. It may happen that the trace embedding theorems are missing. Our proposed interpretation requires to have defined an extension operator , where X is the relevant function space of functions defined on . We show that the proposed interpretation does not depend on the choice of the extension operator . Moreover, we apply our result to obtain existence and uniqueness of solutions to an example of the boundary value problem involving degenerated p-Laplacian with nonhomogeneous boundary condition. Existence of the extension operator within the class of weights when is certain weighted Slobodetskii space and is a Lipschitz boundary domain is confirmed by our previous results.

Higher-order continua derived from discrete media: continualisation aspects and boundary conditions

In this paper, the derivation of higher-order continuum models from a discrete medium is addressed, with the following aims: (i) for a given discrete model and a given coupling of discrete and continuum degrees of freedom, the continuum should be defined uniquely, (ii) the continuum is isotropic, and (iii) boundary conditions are derived consistently with the energy functional and the equations of motion of the continuum. Firstly, a comparison is made between two continualisation methods, namely based on the equations of motion and on the energy functional. They are shown to give identical results. Secondly, the issue of isotropy is addressed. A new approach is developed in which two, rather than one, layers of neighbouring particles are considered. Finally, the formulation and interpretation of boundary conditions is treated. By means of the Hamilton–Ostrogradsky principle, boundary conditions are derived that are consistent with the energy functional and the equations of motion. A relation between standard stresses and higher-order stresses is derived and used to make an interpretation of the higher-order stresses. An additional result of this study is the non-uniqueness of the higher-order contributions to the energy.

BOUNDARY CONDITIONS IN SHALLOW WATER MODELS—AN ALTERNATIVE IMPLEMENTATION FOR FINITE ELEMENT CODES

Finite element solution of the shallow water wave equations has found increasing use by researchers and practitioners in the modelling of oceans and coastal areas. Wave equation models, most of which use equal-orderC0 interpolants for both the velocity and the surface elevation, do not introduce spurious oscillation modes, hence avoiding the need for artificial or numerical damping. An important question for both primitive equation and wave equation models is the interpretation of boundary conditions. Analysis of the characteristics of the governing equations shows that for most geophysical flows a single condition at each boundary is sufficient, yet there is not a consensus in the literature as to what that boundary condition must be or how it should be implemented in a finite element code. Traditionally (partly because of limited data), surface elevation is specified at open ocean boundaries while the normal flux is specified as zero at land boundaries. In most finite element wave equation models both of these boundary conditions are implemented as essential conditions. Our recent work focuses on alternative ways to numerically implement normal flow boundary conditions with an eye towards improving the mass-conserving properties of wave equation models. A unique finite element formulation using generalized functions demonstrates that boundary conditions should be implemented by treating normal fluxes as natural conditions with the flux interpreted as external to the computational domain. Results from extensive numerical experiments show that the scheme does conserve mass for all parameter values. Furthermore, convergence studies demonstrate that the algorithm is consistent, as residual errors at the boundary diminish as the grid is refined.

Local fracture of a composite with linear rigid inclusion

Under the conditions of the plane problem, we propose a new interpretation of boundary conditions in the model problem of elastoplastic equilibrium of a body with linear rigid inclusion whose rupture strength is finite. Stresses in the composite material do not exceed their ultimate values for the materials of the matrix, inclusion, and intermediate contact layer. We studied the two most probable mechanisms of fracture: by exfoliation, i.e., as a result of the propagation of a slip crack along the matrix-fiber interface, and by the rupture of fibers. We established the critical length of the fiber as a function of elastic and strength characteristics of the composite material. If the length of the fiber is greater than critical, the fiber ruptures into two parts; otherwise, the inclusion separates by exfoliation. For each mechanism of local fracture, we determined the ultimate values of external loading.