Modifications of Series Expansions for General End Conditions and Corner Singularities on the Semi-Infinite Strip

1990 ◽  
Vol 57 (3) ◽  
pp. 581-588 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Stiffness Matrix ◽  
Harmonic Wave ◽  
Infinite Cylinder ◽  
Series Expansions ◽  
Infinite Strip ◽  
Corner Singularities ◽  
Numerical Examples ◽  
Time Harmonic ◽  
End Conditions ◽  
Expansion Coefficients

Modified series expansions are used to study semi-infinite isotropic elastic strip problems for general end conditions and corner singularities. The solutions of strips with mixed lateral edges are used as the expansion sets of the end displacement and stress, and an end stiffness matrix, the relation of harmonics of the end displacement and stress, is formed. The present end stiffness matrix approach, an extension to static strip problems of the method by Kim and Steele (1989, 1990) for time-harmonic wave propagation in a semi-infinite cylinder, is effective due to the asymptotic behavior of the stiffness matrix. Also presented is a technique for handling the corner singularities, which is based on the asymptotic analysis of the expansion coefficients of the end stresses. With this, the order and strength of the singularities are determined, local oscillations are virtually suppressed, and converging solutions are obtained. Some numerical examples are given to demonstrate the effectiveness of the approach.

Static Axisymmetric End Problems in Semi-infinite and Finite Solid Cylinders

1992 ◽  
Vol 59 (1) ◽  
pp. 69-76 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Asymptotic Analysis ◽  
Stiffness Matrix ◽  
Present Approach ◽  
Infinite Strip ◽  
Matrix Approach ◽  
Corner Singularities ◽  
End Conditions ◽  
Finite Cylinders

Our earlier technique for a semi-infinite strip (Kim and Steele, 1990) is extended to study general end problems and corner singularities for semi-infinite and finite solid cylinders with free walls. For handling general end conditions, we expand the displacement and stress in term of the Dini series which are the solutions of the cylinders with mixed wall conditions. The relation between the harmonic coefficients of the end displacement and stress is then formed, which we call the end stiffness matrix. One advantage of the end stiffness matrix approach is that the procedure for finite cylinders can be easily built up from that of semi-infinite cylinders. For some end conditions which may yield singular stresses, the nature of the singularity is investigated by the asymptotic analysis of the Dini series coefficients of the stresses. The problems studied by Benthem and Minderhoud (1972) and Robert and Keer (1987) are solved with the present approach.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Frédérique Le Louër ◽  
María-Luisa Rapún
Keyword(s):  
Free Space ◽  
Order Term ◽  
Harmonic Wave ◽  
Dirichlet Condition ◽  
Point Sources ◽  
Neumann Condition ◽  
Scattering Problems ◽  
Content Type ◽  
Time Harmonic ◽  
Topological Gradient

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

scholarly journals Detection of multiple cracks in a beam by means natural frequencies of transverse vibrations

2020 ◽  
pp. 19-26
Author(s):  
Иван Михайлович Лебедев ◽  
Ефим Ильич Шифрин
Keyword(s):  
Arbitrary Number ◽  
Natural Frequencies ◽  
Multiple Cracks ◽  
Transverse Cracks ◽  
Numerical Examples ◽  
Transverse Vibrations ◽  
End Conditions

Рассматривается задача обнаружения множественных, поперечных трещин в стержне с помощью собственных частот поперечных колебаний. В недавней статье авторов доказано, что любое количество трещин однозначно восстанавливается по трем спектрам, отвечающим трем различным типам краевых условий. В статье также предложен алгоритм идентификации повреждений, вносимых трещинами. Помимо этого, высказано предположение, что для однозначной идентификации трещиноподобных дефектов на самом деле достаточно знать два спектра. Для проверки этого предположения разработана модификация предложенного ранее численного алгоритма. Рассмотрены численные примеры. Полученные результаты дают основание полагать, что высказанное предположение справедливо. A problem of detection of multiple transverse cracks in a beam by means of natural frequencies of transverse vibrations is considered. It is proved in the recent paper of the authors that an arbitrary number of cracks can be uniquely determined by three spectra corresponding to three types of the end conditions. An algorithm of reconstruction the damages corresponding the cracks is also developed. In addition, it was assumed that the cracks can be detected using only two spectra. To verify this supposition a modification of the previously developed algorithm is proposed. Numerical examples are considered. The obtained results confirm the assumption.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Effects of Lateral Surface Conditions in Time-Harmonic Nonsymmetric Wave Propagation in a Cylinder

1989 ◽  
Vol 56 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Lateral Surface ◽  
Present Method ◽  
Lateral Wall ◽  
Computational Effort ◽  
Wave Solutions ◽  
Time Harmonic ◽  
End Conditions ◽  
Analogous Manner ◽  
Method Of Solution ◽  
Free Case

The present work is a part of the effort toward the development of an efficient method of solution to handle general nonsymmetric time-harmonic end conditions in a cylinder with a traction-free lateral surface. Previously, Kim and Steele (1989a) develop an approach for the general axisymmetric case, which utilizes the well-known uncoupled wave solutions for a mixed lateral wall condition. For the case of a traction-free lateral wall, the uncoupled wave solutions provide: (1) a convenient set of basis functions and (2) approximations for the relation between end stress and displacement which are asymptotically valid for high mode index numbers. The decay rate with the distance from the end is, however, highly dependent on the lateral wall conditions. The present objective was to demonstrate that the uncoupled solutions of the nonsymmetric waves discussed by Kim (1989), which satisfy certain mixed lateral wall conditions, can be utilized in an analogous manner for the asymptotic analysis of the traction-free case. Results for the end displacement/stress due to various end conditions, computed by the present method and by a more standard collocation method, were compared. The present method was found to reduce the computational effort by orders of magnitude.

ACCURATE FINITE DIFFERENCE REPRESENTATION OF NEUMANN BOUNDARY DATA FOR TIME-HARMONIC WAVE PROPAGATION

1996 ◽  
Vol 04 (04) ◽  
pp. 425-432 ◽  
Author(s):  
ISAAC HARARI
Keyword(s):  
Finite Difference ◽  
Boundary Data ◽  
Truncation Error ◽  
Harmonic Wave ◽  
Neumann Boundary ◽  
Local Truncation Error ◽  
Matrix Equations ◽  
Order Accuracy ◽  
Time Harmonic ◽  
Global Accuracy

Finite difference stencils for inhomogeneous Neumann boundary conditions in acoustic problems with arbitrary source distributions are constructed and analyzed. Boundary stencils are compatible with corresponding interior stencils, preserving symmetry of matrix equations without degrading global accuracy. Higher-order accuracy is attained within the compact support of lower-order methods. Results are verified by local truncation error analysis.

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