Igor Patlashenko

High-order boundary conditions and finite elements for infinite domains

A finite element method for the solution of linear elliptic problems in infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-infinite strip, are considered in detail. In the proposed method, an artificial boundary B is first introduced, to make the computational domain Ω finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B. This condition is localized, and a sequence of local boundary conditions on B, of increasing order, is obtained. The problem in Ω, with a localized DtN boundary condition on B, is then solved using the finite element method. The numerical stability of the scheme is discussed. A hierarchy of special conforming finite elements is developed and used in the layer adjacent to B, in conjunction with the local high-order boundary condition applied on B. An error analysis is given for both nonlocal and local boundary conditions. Numerical experiments are presented to demonstrate the performance of the method.

Discrete Dirichlet-to-Neumann maps for unbounded domains

Abstract It is shown how to construct a discrete counterpart of the Dirichlet-to-Neumann (DtN) map for use on an artificial boundary B introduced in exterior boundary value problems, following the idea of Deakin and Dryden. This discrete map provides an approximate non-reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain Ω bounded by B . Different discrete maps are constructed for use with finite difference and finite element methods in Ω. The solution of some simple problems shows that use of the discrete Dirichlet-to-Neumann (DDtN) map yields nearly the same accuracy as that of using the continuous DtN map.

Optimal local non-reflecting boundary conditions

Abstract A class of numerical methods to solve problems in unbounded domains is based on truncating the infinite domain via an artificial boundary β and applying some boundary condition on β, which is called a Non-Reflecting Boundary Condition (NRBC). In this paper a systematic way to derive optimal local NRBCs of given order is developed in various configurations. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition for C ∞ functions in the L 2 norm. The optimal NRBC may be of low order but still represent high-order modes in the solution. It is shown that the previously derived localized DtN conditions are special cases of the new optimal conditions. The performance of the first-order optimal NRBC is demonstrated via numerical examples, in conjunction with the finite element method.

Non-Reflecting Finite Element Schemes for Three-Dimensional Acoustic Waves

The finite element solution of problems involving three-dimensional acoustic waves in an infinite wave guide, and in the infinite medium around a structure is considered. Such problems are typical in structural acoustics, and this paper concentrates on the efficient numerical treatment of the infinite acoustic medium away from the structure. The unbounded domain is truncated by means of an artificial boundary ℬ. On ℬ, non-reflecting boundary conditions are used; these are either nonlocal Dirichlet-to-Neumann conditions, or their localized counterparts. For the high-order localized conditions, special three-dimensional finite elements are constructed for use in the layer adjacent to ℬ. The performance of the nonlocal and localized boundary conditions is compared via numerical experiments involving a three-dimensional wave guide.

Postbuckling of infinite length cylindrical panels under combined thermal and pressure loading

Abstract An analysis for the pre- and postbuckling behavior of infinite length panels, stiffened and unstiffened, including transverse shear and temperature dependent elastic moduli and thermal expansion coefficients, is proposed. The analysis involves the solution of a system of nonlinear differential equations by means of a relatively new spline-collocation method. Results, yielded by the present study and describing the response of lightly to moderately stringer-stiffened panels and unstiffened ones, subjected to heating and pressure loading, are presented. It is shown that neglect of transverse shear and temperature effects may lead to erroneous results and conclusions.

Non‐local and local artificial boundary conditions for two‐dimensional flow in an infinite channel

The numerical solution of problems involving two‐dimensional flow in an infinite or a semi‐infinite channel is considered. Beyond a certain finite region, where the flow and geometry may be general, a “tail” region is assumed where the flow is potential and the channel is uniform. This situation is typical in many cases of fluid‐structure interaction and flow around obstacles in a channel. The unbounded domain is truncated by means of an artificial boundary B, which separates between the finite computational domain and the “tail.” On B, special boundary conditions are devised. In the finite computational domain, the problem is solved using a finite element scheme. Both non‐local and local artificial boundary conditions are considered on B, and their performance is compared via numerical examples.

Optimal Local Artificial Boundary Conditions

One of the methods commonly used to numerically solve a problem in an infinite domain is the method of artificial boundary conditions [1]. For a linear scalar problem, this method may be summarized as follows: