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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 153

Improvement of the Performance of the Wave Based Method for the Steady-State Dynamic Prediction of Structural Bending Problems

C. Vanmaele, W. Desmet and D. Vandepitte

Department of Mechanical Engineering, Katholieke Universiteit Leuven, Belgium

Full Bibliographic Reference for this paper
C. Vanmaele, W. Desmet, D. Vandepitte, "Improvement of the Performance of the Wave Based Method for the Steady-State Dynamic Prediction of Structural Bending Problems", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 153, 2006. doi:10.4203/ccp.83.153
Keywords: structural dynamics, indirect Trefftz method, Kirchhoff theory, stress singularities.

Summary
In recent years, the vibro-acoustic behaviour of a product has become a design criterion of growing importance. This behaviour is mainly determined by the steady-state dynamic deformations in the mechanical structure of the product. The finite element method (FEM) is commonly used to predict the dynamic behaviour of structures. This method expands the dynamic field variables, within each element, in terms of local, non-exact shape functions. As a result, the size of the model becomes prohibitively large for increasing frequencies, thereby leading to a practical frequency limit. Recently, a wave based prediction technique (WBM), based on the indirect Trefftz method [1], has been developed. Opposed to the FEM, the problem domain is not divided into elements. The field variables are expressed in terms of global wave function expansions, which exactly satisfy the governing dynamic equations. This results in a small model size, and consequently in a high convergence rate compared with the FEM.

The increased convergence rate of the WBM has been demonstrated for various validation examples [2]. Nevertheless, it appears that the WBM suffers from convergence problems when singularities arise in the problem solution [3]. In general it is difficult to achieve accurate prediction results whenever singularities are present. Also for standard FE calculations it is difficult to reach reliable results due to a large pollution error invading the whole domain [4,5]. For a polygonal plate domain, singularities can be expected in the corner points if the interior angle exceeds a critical value. The critical angle depends on the type of boundary conditions along the edges adjacent to the corner. To solve the convergence problems of the WBM for polygonal plate domains, this paper suggests to extend the conventional set of wave functions with a number of corner functions that are capable of representing accurately the singularities. The corner functions are defined as the homogeneous solutions for an infinite wedge domain. These solutions need to satisfy both the homogeneous, dynamic equation and the boundary conditions along the two edges forming the corner. In this way the corner functions represent the exact state of stress that is expected in the corner of the plate domain. However, it is only possible to define the exact solution if the two radial edges are both simply-supported. For other combinations of radial boundary conditions, a procedure is presented to derive the corner functions from Williams' static eigenfunctions [6], such that the singular behaviour in the vicinity of the corner is asymptotically correct.

The capabilities of this enhanced WBM are demonstrated by three validation examples. A first example consists of a clamped plate. In this case no singularities are present and the WBM does not have to be adapted. It is shown that the WBM does not suffer from dispersion errors and that it is capable of making accurate predictions with a substantially smaller computational effort as compared to the FEM. The second example consists of a simply-supported plate. In this case singularities are present. Therefore the WBM is extended with some additional corner functions. Since all boundaries are simply-supported the exact analytical solutions can be used as corner functions. The enhanced WBM no longer suffers from convergence problems and is shown to have a higher convergence rate than the FEM. Also for the last example that consists of two clamped and two free boundaries, singularities are present. In this case, the corner functions must be derived from the static eigenfunctions. It is demonstrated that also with the derived corner functions the convergence problems are solved and a high convergence rate is achieved. As a conclusion it may be stated that the enhanced WBM is capable of making accurate predictions, even if the problem solution contains singularities. Moreover, it shows to have an increased convergence rate compared to the FEM. This computational efficiency allows the WBM to be applied towards higher frequencies and relaxes the frequency limitation of the FEM.

References
1
E. Trefftz, "Ein Gegenstück zum Ritschen Verfahren", in "Proceedings of the 2nd International Congress on Applied Mechanics", pages 131-137, Zürich, Switzerland, 1926.
2
W. Desmet, "A Wave based prediction technique for coupled vibro-acoustic analysis", Ph.D. dissertation, Katholieke Universiteit Leuven, Departement Werktuigkunde, Leuven, 1998.
3
C. Vanmaele, W. Desmet, D. Vandepitte, "On the use of the wave based method for the steady-state dynamic analysis of three-dimensional plate assemblies", in "Proceedings of ISMA2004", Leuven, 2004.
4
J. Robinson, "An evaluation of skew sensitivity of thirty three plate bending elements in nineteen FEM systems", Finite Element News - special report, 1985.
5
R.H. MacNeal, "Finite Elements: their design and performance, Marcel Dekker Inc, New York, 1994.
6
M.L. Williams, "Surface Stress Singularities Resulting from various Boundary Conditions in Angular Plates under Bending", in "Proceedings of the First U.S. National Congress of Applied Mechanics",325-329,1952.

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