The alpha interpolation method for the solution of an eigenvalue problem

Niki, Hiroshi; Sawami, Hideo; Ikeuchi, Masatoshi; Okamoto, Naohisa

doi:10.1016/0771-050x(82)90002-x

Cited by 4 publications

(13 citation statements)

References 5 publications

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“…The α interpolation of FEM and FDM on a rectangular domain discretized by structured simplicial FE mesh would yield a scheme identical to the AIM wherein the mass matrix that appears in the Galerkin FEM is replaced by an α ‐interpolated mass matrix. In the current PG method, we recover the AIM (even on unstructured simplicial meshes) by making the choice

{\overset{W}{false}}^{a} MathClass-rel= N^{a} {MathClass-rel|}_{{scriptE}_{h}}

.…”

Section: Discussionmentioning

confidence: 99%

“…Nevertheless, just like for the GLS-FEM, the coefficient a 0 can be chosen so as to arrive at a higher-order modification of the interior stencil of the Galerkin FEM. Similar studies for eigenvalue problems using the AIM with simplicial FEs was carried out in [4,5,19,20].…”

Section: Simplicial Finite Elementsmentioning

confidence: 93%

“…We see that by using simplicial FEs in 2D, the α interpolation of Galerkin FEM and FDM is equivalent to the AIM . In the AIM, the consistent mass matrix M that appears in the Galerkin FEM is replaced by the α ‐interpolated mass matrix M α := (1 − α ) M + α M L .…”

Section: Simplicial Finite Elementsmentioning

confidence: 99%

“…We see that by using simplicial FEs in 2D, the˛interpolation of Galerkin FEM and FDM is equivalent to the AIM [4,5]. In the AIM, the consistent mass matrix M that appears in the Galerkin Consider the BVP (1) posed on an interior 2D domain subjected to Dirichlet boundary conditions and let f .x/ D 0.…”

Section: Simplicial Finite Elementsmentioning

confidence: 99%

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A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Nadukandi

Oñate

García

2011

Numerical Meth Engineering

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SUMMARYA new Petrov-Galerkin (PG) method involving two parameters, namely˛1 and˛2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is,˛1 D˛2 D˛; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18-46) for the Helmholtz equation if the parameters are distinct, that is,˛1 ¤˛2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by 1 ) and the mass terms (specified by˛2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18-46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions.

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{\overset{W}{false}}^{a} MathClass-rel= N^{a} {MathClass-rel|}_{{scriptE}_{h}}

.…”

Section: Discussionmentioning

confidence: 99%

Section: Simplicial Finite Elementsmentioning

confidence: 93%

Section: Simplicial Finite Elementsmentioning

confidence: 99%

Section: Simplicial Finite Elementsmentioning

confidence: 99%

See 2 more Smart Citations

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Nadukandi

Oñate

García

2011

Numerical Meth Engineering

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“…M α : = α M + (1 − α) M L . This GMM scheme was later baptized as the alpha‐interpolation method (AIM) 62 and was extended to the hollow waveguide analysis in 63 and the Schrodinger equation in 64. For the simple 1D case, our scheme mimics the AIM and in 2D making the choice α = 0.5 we recover the generalized fourth‐order compact Padé approximation 38, 39 (therein using the parameter γ = 2).…”

Section: Introductionmentioning

confidence: 99%

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

Nadukandi

Oñate

García

2010

Numerical Meth Engineering

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We propose a fourth-order compact scheme on structured meshes for the Helmholtz equation given by R():= f (x)+D+n 2 = 0. The scheme consists of taking the alpha-interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha-interpolation method (J. Comput. Appl. Math. 1982; 8(1):15-19) and in 2D making the choice = 0.5 we recover the generalized fourth-order compact Padé approximation (J. 128:325-359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325-359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((n ) 4 ), where n, represent the wavenumber and the mesh size, respectively. An expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L 2 norm, the H 1 semi-norm and the l ∞ Euclidean norm are done and the pollution effect is found to be small. 19As the current problem admits a variational principle, naturally, discretization methods based on variational formulations viz. the Galerkin and the Trefftz-Galerkin-type methods have been preferred to other methods. The Galerkin-type methods are domain-based wherein the integral statement involves only the weak form of the governing differential equation and the sub-space of test-functions are assumed to satisfy a priori the kinematic compatibility and essential boundary conditions. The Trefftz-Galerkin-type methods are boundary-based and are formulated using the reciprocal principle wherein the integral statement involves only the kinematic compatibility and essential boundary conditions of the problem and the sub-space of test-functions are assumed to satisfy a priori the governing differential equation [1,2].In the context of the Galerkin-type methods, the finite element method (FEM) is a powerful technique to systematically generate subspaces of test-functions (classically piecewise polynomial spaces). Some of the earlier works on the use of FEM for the numerical solution of the Helmholtz equation can be found in [3-10] and the references cited therein. In [5,7] error estimates were given for the asymptotic (n 2 assumed sufficiently small) and pre-asymptotic (n assumed sufficiently small) cases, respectively. It was shown that for the discrete problem the LBB ‡ constant can be expressed as h = min{| h m −n 2 |/ h m } [6]. Thus, for the continuous problem (visualized as → 0) the LBB constant can be expressed as = min{| m −n 2 |/ m }, which in an average sense implies that is inversely proportional to the wavenumber n, i.e. ∝ n −1 [6,7]. Thus for high wavenumbers and for the case of degeneracy (n → h m ), the LBB constant for the discrete problem tends to be small which in turn leads to a loss of stability. The loss of stability...

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Angular distribution of energy release rates and fracture of piezoelectric solids

Rajapakse

2004

Smart Mater. Struct.

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Applicability of energy release rate criteria to piezoelectric materials is examined in this paper. The angular distribution of strain and total energy release rates at a crack tip in a plane piezoelectric medium is first formulated by using a branched crack model. By relaxing the commonly held assumption of self-similar crack propagation and incorporating in the fracture toughness anisotropy, the criteria of modified strain energy release rate and modified total energy release rate are applied to discuss the propagation path of an impermeable crack. Numerical results indicate that the predictions based on the criterion of modified total energy release rate do not show qualitative agreement with experimental observations, whereas the predictions based on the criterion of modified strain energy release rate appear more promising. Based on the criterion of modified strain energy release rate, a crack tends to branch off from a straight path under a positive electric loading, regardless of crack orientation. An applied electric field can either promote or retard crack propagation depending on the direction of electric field, crack orientation and crack extension direction. The predictions made by the energy-based criteria are compared with those corresponding to a recently reported stress-based criterion.

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The alpha interpolation method for the solution of an eigenvalue problem

Cited by 4 publications

References 5 publications

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

Angular distribution of energy release rates and fracture of piezoelectric solids

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