Finite element formulations for exterior problems: application to hybrid methods, non‐reflecting boundary conditions, and infinite elements

Numerical Meth Engineering

2016

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SUMMARY In this work, we use Nitsche’s formulation to weakly enforce kinematic constraints at an embedded interface in Helmholtz problems. Allowing embedded interfaces in a mesh provides significant ease for discretization, especially when material interfaces have complex geometries. We provide analytical results that establish the well-posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche’s method is used to enforce kinematic constraints. As in the analysis of conventional Helmholtz problems, we show that the inf-sup constant remains positive provided that the Nitsche’s stabilization parameter is judiciously chosen. We then apply our formulation to several 2D plane-wave examples that confirm our analytical findings. Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis.

Section: Homogeneous Domain We Consider a Two Dimensional Version Ofmentioning

confidence: 99%

Nitsche's method for Helmholtz problems with embedded interfaces

Zou

Aquino

Numerical Meth Engineering

2016

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“…This functional is based on a complex-valued mixed formulation in which continuity across the artiÿcial boundary is enforced by Lagrange multipliers. 21 The Euler-Lagrange equations are employed to eliminate the Lagrange multipliers and obtain the form shown here. Continuity is enforced by energy ux-type terms, reminiscent of the radiation condition that is being replaced.…”

Section: Variational Statement With Weak Continuitymentioning

confidence: 99%

“…However, in support of the plausibility of the procedure proposed herein, we note that representing the outer ÿeld in terms of global circumferential harmonics leads to a new derivation of the known DtN method, 11 for which stability in the BabuÄ ska-Brezzi sense is not an issue for any ÿnite element interpolation of the inner ÿeld. 21 The weighting functions in the weak form (19) are not conjugated, in accordance with the complex-valued functional (10). A similar weak form with conjugated weighting functions, as employed by the wave envelope procedure, 16 as employed by the wave envelope procedure, 16 can be derived directly from the Euler-Lagrange equations (14) - (17).…”

Section: Variational Statement With Weak Continuitymentioning

confidence: 99%

“…The functional is deÿned on the bounded inner domain and interface only, motivating the term boundary inÿnite elements. This formulation is exact, free of constraint degrees of freedom, and can be used to recover many well-known formulations developed previously, 21 including the DtN boundary conditions. A related formulation that weakly enforces boundary conditions was introduced in the context of elastostatics.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Boundary infinite elements for the Helmholtz equation in exterior domains

Int. J. Numer. Meth. Engng.

Barbone

Slavutin

et al. 1998

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A novel approach to the development of inÿnite element formulations for exterior problems of time-harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domainbased computation of exterior problems. Special cases include non-re ecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across inÿnite element boundaries, typical of standard inÿnite element approximations. Continuity between ÿnite elements and inÿnite elements is enforced weakly, precluding compatibility requirements. Various inÿnite element approximations for two-dimensional conÿgurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme. ?

“…In contrast to hybrid methods, Nitsche's formulation is free of auxiliary fields, simplifying the theory and reducing computational cost. Similar ideas were used to enforce weak matching for exterior problems of acoustics [25]. Variational consistency provides robust performance with respect to the value of the method parameters, as opposed to traditional penalty schemes.…”

mentioning

confidence: 99%

A unified approach for embedded boundary conditions for fourth‐order elliptic problems

Numerical Meth Engineering

Grosu

2014

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An efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier-Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche's approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non-conforming meshes with non-interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B-splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter. 656 I. HARARI AND E. GROSU of freedom. However, the method suffers from the lack of a rigorous criterion for defining its stabilization parameter. A discontinuous Galerkin method [9] alleviates this difficulty. Another approach [10], with deflection and rotation degrees of freedom, is based on a stabilized method for a mixed representation.Smooth functions such as B-splines attain higher global continuity with relative ease, making them suitable for higher-order problems. B-splines of degree k are typically C k 1 continuous. Such bases are becoming increasingly popular for use in finite element computations, in part due to their ability to provide exact geometric representations (as with the isogeometric concept [11]). This has led to renewed interest in direct discretization of C 1 theories such as thin plate (and shell) bending [12][13][14].Conventional finite elements based on virtual work treat kinematic boundary conditions as essential conditions, requiring the basis to be interpolatory and the mesh to conform to the boundary of the domain. Weakly enforcing the kinematic boundary conditions eliminates the geometric limitation of classical C 1 elements such as Bogner-Fox-Schmit. This notion was explored in the context of hybrid methods [15,16]. (Similar ideas are used to enforce inter-element continuity for timeharmonic waves [17].) However, hybrid methods require additional degrees of freedom for the auxiliary fields and, as in the case of mixed methods, are subject to stability issues.We present a variational formulation for fourth-order elliptic problems with embedded Dirichlet boundary conditions based on the Nitsche approach. This is one of several techniques under the broad category of immersed boundary and interface...