Transactions on Modelling and Simulation vol 7, 1994 WIT Press, ISSN X

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1 On the calculation of natural frequencies of microstructures using the DRBEM T.R. Bridges & L.C. Wrobel Wessex Institute of Technology, University of Portsmouth, Ashurst Lodge, Ashurst, Southampton, S040 7AA, UK 1 Introduction This paper discusses the calculation of the natural frequencies of thin structural components using the Dual Reciprocity boundary element method. This technique, introduced by Nardini and Brebbia in 1982, has certain advantages over other boundary element techniques in that it produces a linear, time independent, boundary only system which is compatible with standard 'black box' eigenvalue solvers. Results are presented which provide comparison with theoretical approximations and show how the method behaves for a more realistic application, a microstructure, for which finite element results are available. Traditional integral equation formulations, involving the elastodynamic fundamental solution, which is complex valued and frequency dependent, lead to a non-linear system [1-4]. One is therefore forced to find the minimum value of the complex determinant of the system of equations which is an extremely inefficient, error prone process. For this reason, it has recently been more popular to use approximate formulations employing the elastostatic fundamental solution which, through its independence of frequency, leads to a linear, real valued system. The domain integral which results from such a formulation, has been treated in different ways as outlined below. The mixed boundary integral-finite element approach [5] requires discretization of the domain using internal cells. Although producing a standard eigenvalue problem, the presence of internal variables generates larger systems of equations. Alternatively, the Multiple Reciprocity Method (MRM) [6] involves a transformation of the domain integral into a series of equivalent boundary integrals by using a sequence of higher order fundamental solutions. This method has been successfully applied to acoustics [7-9] i.e. the Helmholtz equation where the series of higher-order fundamental solutions are complete and convergent. However, for elastodynamics, the series is not complete and may converge to an incorrect solution. Another possibility is to use the Dual Reciprocity Method (DRM) [10], which approximates the inertia term using global interpolation functions as will be explained in more detail in the following section. Again, a standard eigenvalue problem results, but without the need for internal discretization [11-15]. The eigenvalue solver used in the present case was taken from Numerical Recipes [16] which implements the Q-R algorithm preceded by reduction of the system matrix to upper Hessenberg form [17]. Other possibilities exist such as the Lanczos algorithm [18] which may increase the efficiency of the solution.

2 530 Boundary Element Method XVI 2 Dynamic Analysis The following description attempts to outline the essential concepts which are needed in the formulation of the elastodynamic problem using dual reciprocity. In order to keep the description uncluttered, no time will be expended on basic boundary element techniques which are well documented in, for example, reference [19]. The equilibrium state of an elastic homogeneous body can be stated as which requires boundary conditions and Transactions on Modelling and Simulation vol 7, 1994 WIT Press, ISSN X initial conditions puk (1) Ui Ui on FI, pi (TijTij --= pi on 1^ (2) %,=w9, 6, = 60 (3) The integral equation statement of the problem can be written as t%t y ^tpt y mt^t py ^^ which has employed the elastostatic fundamental solution w*&, namely Kelvin's solution of the Navier equation C«*,M + j-r^;wt,m + A<ei = 0 (5) where A^ is the Dirac delta and ej are orthogonal unit vectors. The effect of the DRM is to reduce the last remaining domain integral over the accelerations i/& in (4) to integrals over F. This is done by introducing the approximation where fi are known approximating functions and o^ are unknown, time dependent coefficients for directions k = 1,2. The functions employed in this instance are simply /' = a + r (7) where a, so as not to dominate r, is the average of all element lengths. Given a particular stress field one can substitute for u in the domain integral and, applying the divergence theorem, eventually obtain (8) r r J=\ r r where itmk is obtained from writing (8) as the Navier equation giving 1-2i/ o 1 =

3 from which it follows that Boundary Element Method XVI 531 y^l_^ [(4-5f/)r,nm - (1-5i/)r^n, + [(4 - Supposing now that Q is discretized using TV boundary nodes and / internal interpolation points, the boundary element equation can now be expressed in terms of matrices as N N N+I If this equation is applied at each node in turn, the system Hu - Gp = p(hu - GP)a (13) is obtained, in which c has been combined with H to produce H. Supposing now, that u(x,y,t) varies harmonically with time, then where u; is angular frequency. Then % = -w2% (14) Hu - Gp = u~p(gp - HU)F-IU = w^su (15) Note that the harmonic behaviour of u removes the time dependency and so now, it is only necessary to apply the boundary conditions (2). On doing this, one can reduce the size of the system by partitioning the matrices according to components which correspond to either I\ or Fo like so: Hoi Hoo U2 Goi G U2 ^ ^ and then eliminating p^ to obtain (H22 - G2lGljHi2)u2 - ^(822 ~ ^ig^s^)^. (17) The above generalized eigenvalue problem can be solved using a library routine capable of dealing with real, general matrices such as the Q-R or Lanczos algorithms. 3 Applications 3.1 Example 1 - One-Dimensional Beam The axial modes of a very thin structural component can be compared with the theoretical solution for a one-dimensional cantilever by making the aspect ratio very large and using Poisson's ratio equal to zero. The model, shown in figure 1, is fixed at one end in both x and y directions and has E p - 1. Several cases were analysed with increasing aspect ratios ranging from 1:10 up to 1:1000. Obviously, the higher the aspect ratio, the closer the approximation to a one-dimensional body, but problems begin to arise when elements get very close to one another since the radial function? approaches zero making the integration nearly singular. This problem was overcome using the integration scheme of Telles [20] in which a cubic transformation is employed to shift the Gauss points towards the near singular point.

4 532 Boundary Element Method XVI Figure 1: Dimensions of the Thin Beam The theoretical solution for the natural frequencies of a one-dimensional beam of length, density p and Young's modulus E, is given by Craig [21] as The bar chart shown in figure 2, displays the percentage error in the first five natural frequencies obtained using only 22 elements. It is interesting to note that some negative eigenvalues were obtained for the higher aspect ratios. These could be made to disappear by improving the integration, either by increasing the number of elements, or the number of Gauss points on singular elements. However, there was no significant change in the positive values. Table 1 shows the magnitude of the percentage error for the beam with ratio 50:1. The first five frequencies compare wall with the analytical approximation. Natural Frequencies for the Ratio 50:1 Natural Frequency BEM Analytical % Diff Wl LJi Wg W W Table 1: Comparison with Analytical Approximation 3.2 Example 2 - Accelerometer Hinge Tests have been carried out which have shown the DRM to be effective in solving problems for which the structural geometry is relatively uncomplicated [22]. This example illustrates how the method behaves for a component which has sections of very different orders of magnitude. The model, shown in figure 3, is part of an accelerometer for use in aircraft navigation systems. It has total length of 3.4 mm and width 0.4 mm. The blade, which is the very narrow section connecting the two end blocks, is a mere 8 /im across and has a length of 0.3 mm which gives it an aspect ratio of 1:37.5. The Young's modulus used was 1.9x10^ N/rnrn^ with density p = 8.93x10"^ Kg/mm? and a Poisson's ratio of The differing orders of dimension of the central section meant that meshing the model was not as straightforward as one might expect. Finding a satisfactory discretization proved to be quite difficult and simply doubling the number of elements did not always mean that an inadequate mesh could be rectified. The second natural frequency and beyond were very stable and convergence was achieved quite rapidly, however, the first

5 Boundary Element Method XVI 533 Errors in Natural Frequencies :1 20:1 50:1 100:1 Ratio a:b 250:1 500:1 1000:1 Figure 2: Percentage Errors in Natural Frequencies for Different Ratios a : 6 frequency was much less stable and was the source of the difficulties. This is unfortunate, since for some applications, it is the first fundamental frequency that is the most important. It should be noted that similar problems were encountered when analysing the model usingfiniteelements, and so this instability is not unique to the DRM. However, results which compared very well with those obtained by finite element analysis were achieved using just 76 elements, around half of these being concentrated around the sensitive area,figure3 (b). Exact details of thefiniteelement mesh are unavailable although it is known that the mesh was refined until satisfactory convergence was achieved. Experimental results were obtained for this structure by gluing it to a piezoceramical bimorph and driving it with a sinus-voltage signal of variable frequency. Displacements of more than 0.5 /im were observed using an optical stereo microscope (x!50) and a scanning electron microscope. Information relating blade thickness to the first fundamental frequency was obtained for a few specimen hinges revealing a linear relationship from which the results given here could be extrapolated. Due to the rounded edges of the structures, the thickness could only be measured to an accuracy of 0.5 pm and the 'bestfit'line through the data points was drawn by eye. Table 2 shows the results relating to a series of meshes starting with 76 elements and doubling each time. Experimental data is only available for the first mode since sufficiently accurate measurements cannot be made for the much higher frequencies of the other modes. 4 Conclusions The dual reciprocity boundary element method is an effective means of calculating the eigenfrequencies of a structural system using a direct approach rather than employing a sweeping frequency analysis. The use of the elastostatic fundamental solution makes

6 534 Boundary Element Method XVI (a) (b) (c) (d) Figure 3: (a) Geometry of the accelerometer hinge with the mesh sensitive area (b). (c) First mode, (d) Second mode. Natural Frequencies of the Accelerometer Hinge DRBEM FEM Experimental Mode Mode Mode Table 2: Comparisons with FEM and experimental results.

7 Boundary Element Method XVI 535 it possible to form a standard eigenvalue system which can then be passed to a "black box" solver. So far, it has been possible to obtain accurate results without the need to approximate the inertia term throughout the domain using internal points. This is useful for ease of modelling and in reducing the number of degrees of freedom. This paper has demonstrated that accurate results can be obtained for problems where the geometry presents difficulties in integration due to the radial function r approaching zero. However, care must be taken when choosing a suitable discretization since an inadequate mesh can lead to seemingly unstable results for low frequencies. The examples presented here are a preliminary part of on-going work investigating different kinds of microstructures, which will include extension to three dimensions. 5 Acknowledgement This research forms part of the European Commission ESPRIT project 6874, Microsystem Analysis and Simulation System (MASS). The accelerometer hinge analysed here was designed by MicroParts. The finite element analysis of this structure was carried out by Dr. A. Larhmann from the Technical University of Berlin and the experimental data was obtained by Dr. C. Lefimollmann of MicroParts. References [1] G. De Mey, "Calculation of eigenvalues of the Helmholtz equation by an integral equation method," WermaZzoW JowrW /or Mfmerzca/ Mef/Ws zn Engzneerzng, Vol. 10, pp 59-66, [2] G.R.C. Tai and R.P. Shaw, "Helmholtz equation eigenvalues and eigenmodes for arbitrary domains," JowrW o/yae /Icowshca/ 5"ocze^ o/amerzca, Vol. 56, pp , [3] Y. Niwa, S. Kobayashiand M. Kitahara, "Determinationof eigenvalues by boundary element methods," in DeWopmemfs zn BowWar%/ E/ememi AWWs, Vol. 2, Applied Science Publishers, London, [4] J.O. Adeyeye, M.J.M. Bernal and K.E. Pitman, "An improved boundary integral equation method for Helmholtz equation," /nfernafzow Jo%rW/or g, Vol. 21, pp , [5] G. Bezine, "A mixed boundary integral-finite element approach to plate vibration problems," Mechanics Research Communications, Vol. 7, pp , [6] A.J. Nowak and C.A. Brebbia, "The multiple reciprocity method. A new approach for transforming BEM domain integrals to the boundary," Engineering Analysis, Vol. 6, pp , [7] N. Kamiya and E. Andoh, "Eigenvalue analysis by boundary element method," Jo^nia/ o/.s'otw and ^razzon, Vol. 160, pp , [8] N. Kamiya and E. Andoh, "Standard eigenvalue analysis by boundary element method," Commwnzca/zmts 2% A^merzcaf Me</Ws z% En^zneermg, Vol. 9, pp , [9] N. Kamiya, E. Andoh and K. Nogae, "Eigenvalue analysis by the boundary element method: new developments," Ez^meerzng /Ima/yszs, Vol. 12, pp , 1993.

8 536 Boundary Element Method XVI [10] P.W. Partridge, C.A. Brebbia and L.C. Wrobel, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton and Elsevier, London, [11] D. Nardini and C.A. Brebbia, "A new approach to free vibration analysis using boundary elements," in Proc. Jth Int. Conf. Bound. Elms., Computational Mechanics Publications, Southampton and Springer- Verlag, Berlin, [12] P.K. Banerjee, S. Ahmad and K.C. Wang, "A new BEM formulation for the acoustic eigenfrequency analysis," International Journal for Numerical Methods in Engineer- ;%?, Vol. 26, pp , [13] J.P. Coyette and K.R. Fyfe, "An improved formulation for acoustic eigenmode extraction from boundary element models," Trans. ASME, Journal of Vibration and Acoustics, Vol. 112, pp , [14] A. Ali, C. Rajakumar and S.M. Yunus, "On the formulation of the acoustic boundary element eigenvalue problems," International Journal for Numerical Methods in f, Vol. 31, pp , [15] C. Rajakumar, A. Ali and S.M. Yunus, "Lanczos algorithm for acoustic boundary element eigenvalue problems," Journal of the Acoustical Society of America, Vol. 91, pp , [16] W. H. Press, W. T. Tenkolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes The Art of Scientific Computing, Cambridge University Press, [17] J. Wilkinson, The Algebraic Eigenvalue Problem, Springer- Verlag, Berlin, [18] C. Rajakumar and C. R. Rogers, "The Lanczos algorithm applied to unsymmetric generalized eigenvalue problems," International Journal of Numerical Methods in Engineering, Vol 32, pp , [19] C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary Element Techniques, Springer- Verlag, Berlin and New York, [20] J. C. F. Telles, "A Self-Adaptive Co-ordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals," International Journal for Mfmenco/ MeMock m #%# meermp, Vol. 24, pp , [21] R. R. Craig, Structural Dynamics, Wiley, Chichester, [22] C. A. Brebbia and D. Nardini, "Dynamic analysis in solid mechanics by an alternative boundary elements procedure," International Journal of Solid Dynamics and Earthquake Engineering, Vol 2, pp , 1983.

Abstract INTRODUCTION

Abstract INTRODUCTION Iterative local minimum search for eigenvalue determination of the Helmholtz equation by boundary element formulation N. Kamiya,E. Andoh, K. Nogae Department ofinformatics and Natural Science, School of