Modeling Accuracy in FEA of Vibrations of a Drumhead

Shock and Vibration, Jul 2018

The study of the problem of predicting values of Rayleigh’s quotient for a square drumhead provides a basis for assessing the relation between grid size, accuracy of analysis results, and efficiency of data processing in finite element analysis. The analysis data indicate that unacceptable grid sampling can occur even for the fine grids, that strictly monotonic convergence is attainable for vibration analysis, and that more efficient computer analysis associates with use of curve fitting analysis of conventional finite element analysis results.

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Modeling Accuracy in FEA of Vibrations of a Drumhead

Shock and Vibration, Vol. Modeling Accuracy in FEA of Vibrations of a Drumhead Robert J. Melosh 0 0 Department of Civil and Environmental Engineering Duke University Durham , NC 27706 , USA The study of the problem of predicting values of Rayleigh's quotient for a square drumhead provides a basis for assessing the relation between grid size, accuracy of analysis results, and efficiency of data processing in finite element analysis. The analysis data indicate that unacceptable grid sampling can occur even for the fine grids, that strictly monotonic convergence is attainable for vibration analysis, and that more efficient computer analysis associates with use of curve fitting analysis of conventional finite element analysis results. © 1993 John Wiley & Sons, Inc. - INTRODUCTION Finite element analysis modeling approximations are the inaccuracies introduced in the numerical model that vanish as the grid interval approaches zero. Modeling approximations include the defi­ nition of the original and deformed geometry of the structure, the equations and coefficients of the material constitutive model, and the state­ ment of the problem boundary conditions. Consider the structural configuration defined by Figure 1. This thin, square-shaped flat surface is pinned along its outer boundary. The uni­ formly thick surface has a uniform mass density. The membrane is stretched over a square frame, resulting in a uniform tensile force in the mem­ brane. We focus on the relation between the grid interval, accuracy, and the number of calcula­ tions required in predicting the Rayleigh quo­ tients (RQ) of the drumhead of Figure 1. Limiting our attention to the linearized equa­ tions of equilibrium and isotropic homogeneous material behavior leads to the differential equa­ tion given by Young (1962): where w is the lateral displacement and x and y are the coordinates along the x and y axes, t is the time variable, JL is the mass density, and N is the tensile force in the membrane. The analytical solution to the square drum­ head problem is expressed by w = Apq sine(p7TxIH) sine(q7TyIH) sine(fpq) (2) where Apq are arbitrary nonzero constants, p and q are positive, nonzero integers, H is the length of a side of the square membrane, and,.t;,q are the resonant frequencies. Thus, the modes are products of half sine waves over the planform of the structure. For a given q, mode displacement is an even function of x when p is odd and an odd function when p is even. Figure 2 shows the location of nodal vibration lines for the first 16 modes of the membrane. The nodal curves are straight lines that are orthogo­ nal to each other. Substitution of Eq. (2) in Eq. (1) prescribes that, where M is the total mass of the surface. Rayleigh's quotient can be evaluated by, -HI2 HI2 x Jl = mass density Section A-A FIGURE 1 Drumhead geometry. that is, Rayleigh's quotient is symmetric in p and q. The approximations made in arriving at the governing field equations are approximations of mathematical idealization. Because the mathe­ matical model is a differential equation, it implies no FEA modeling approximations. Converting the mathematical model to a finite number of equations of motion does incur FEA THE PROBLEM: MODELING IN FEA OF A VIBRATING DRUMHEAD w=Apq sine(pxxlH) sine(qR}'/H) sine(Bt) G-B p=1, q=1 p=1, q=2 § p=1, q=3 p=3,q=1 ITDffij~~ p=3,q=2 p=3,q=3 p=3,q=4 DTIJtHE§E p=4,q=1 p=4,q=2 p=4,q=3 p=4,q=4 modeling approximations. The characteristics of the approximations are determined by the finite element model, the shapes of the element, alloca­ tion of the grid lines and nodes over the geometry of the structure and refinements of the finite ele­ ment grid. In the next section of this paper, we define the FEA model used in the experiments in more de­ tail. Then the results of conventional FEA of the drumhead are displayed. The basis and results of adding curve fitting analysis to conventional analysis are presented. The sensitivity of analy­ sis results as a function of the required accuracy and the number of vibration modes are esti­ mated. Converting the mathematical model to a finite number of equations of motion incurs FEA mod­ eling approximations. The characteristics of the approximations are determined by the finite ele­ ment model, the shapes of the element, alloca­ tion of the grid lines and nodes over the geometry of the structure and refinements of the finite ele­ ment grid. We choose the following finite element model­ ing components: 1. The element model is based on hyperbolic variation of lateral displacement with the x and y coordinates. Thus, displacements have CO continuity across element bound­ aries. 2. The energy of displacements is calculated by integrating the energy density sampled at nodes of a 3 x 3 network of Gauss inte­ gration points. Then, because the displace­ ment function is a polynomial, the quadra­ ture yields the values of the strain and kinetic energies without additional model­ ing approximations. Rayleigh's quotients for the structure are de­ fined by, RQpq = KE/SE = 0.5 Lw™wl(O.5 LwTMw) (5) where KE is the kinetic energy, SE is the strain energy, w is a normalized column eigenvector of lateral displacements, K is the symmetric fourthorder rank two element positive semidefinite stiffness matrix, M is the symmetric fourth-order rank four element positive definite mass matrix, and the summation, ~, extends over all elements of the finite element model. Each gridwork of elements constitutes a regu­ lar isogeometric subdivision of the original one by one grid of the drumhead. The sequence of finite element grids involves relative grid inter­ vals, hIH, of 1/1, 112, 1/3, 1/4, 1/5 . . . 1/19. The objective of the analysis is to evaluate Rayleigh's quotient to 1.3 digits of accuracy us­ ing a finite element model. This accuracy corre­ sponds to tolerance of about 5% error in analysis results: the accuracy aspired to in engineering practice. For research purposes all calculations are per­ formed using IEEE double precision arithmetic (about 15.6 digits of precision). The double-preci­ sion results of the conventional finite element analysis are truncated to single precision (6.92 digits of precision) for curve fitting analysis. The maximum loss of precision measured for all fit­ tings was two digits. Therefore, the maximum number of digits of accuracy in the computer results is 4.9 digits. The number of digits of accuracy is measured by, DA = sygnum (XI - X2) 10glO [abs(xl - x2)1 (0.5(xl + X2)] (6) where DA is the number of decimal digits of ac­ curacy, sygnum(. . .) takes the sign ofa number, and, XI and X2 are the values of the two numbers . being compared. This definition implements the comparison of two independent estimates of a variable, X, without bias in the selection of the more meaningful estimate. In this paper, the XI are values of Rayleigh's quotient given by Eq. (3) and X2 are the values of Rayleigh's quotient eval­ uated by FEA. We measure the data processing resources needed for analysis by, CI = ~i(Hlhi)2, with i = 1,2, . . . NM, and NM = q * (q + 1)/2 (7) where CI is the calculations index, NM is the number of modes required by the analyst, and, q ;::: p, or the roles of p and q are reversed. This definition implies that the resources associated with curve fitting analysis are negligible com­ pared with those required in generating element stiffness and mass matrices and evaluating ener­ gies. Furthermore, we assume that the number of modes required by the analyst starts with mode (1, 1) and proceeds to mode (q, q) by ordering by increasing (p2 + q2). A SURVEY OF THE FEA SOLUTION SPACE Table 1 lists the accuracy of 190 finite element solutions for 19 grids and 10 vibration modes. Because Rayleigh's quotient is symmetric in p and q, the data of Table 1 are pertinent to analy­ sis of the lowest 16 vibration modes and 19 grids and hence, 304 solutions. Table 1 cells with more than a decimal number indicate spurious FEA estimates of quotient val­ ues. Cells containing 0/0 are cases for which both the strain energy and the kinetic energy are zero because each FEA node is located on a vibration nodal line. Cells that discriminate more modes than the number of active degrees of freedom have linearly dependent modes that violate the orthogonality conditions. Defining the number of active degrees of free­ .dom as the number of generalized nodal displace­ ments that take on nonzero values of displace­ ments, we observe that linearly dependent modes are excluded when the FEA gridwork sat­ isfies the criteria, Hlhe > max(q, p) (8) where Hlhe is the coarseness limit on acceptable grid intervals, and max(p, q) indicates the maxi­ mum ofp or q. The sign of the decimal number in each ac­ ceptable cell is positive when the FEA produces quotient estimates that are greater than the value given by Eq. (3). For q ;::: p, these involve values of HIh such that, Hlh = 1 * q, 2 * q, 3 * q . (9) Figure 3 displays the relation between accu­ racy and grid interval for modes (1, 1), (1, 2), and (4, 4) developed by plotting results of the analy­ ses of Table 1. These graphs portray the S­ shaped convergence curves that are common in FEA applications (Melosh, 1990) . Figure 3 shows two curves for mode (1, 2): one for the cells of Table 1 that involve positive DAs and one for cells with negative digits. Sepa­ rating the data in this way results in two monotonically converging sequences instead of one curve with an oscillating sequence of quotient accuracy values. The convergence curve for mode (1, 1) also appears in Figure 3. This curve illustrates strictly monotonic convergence associates with mode (1, 1), that is,successively lower relative grid intervals result in successively higher values of DA and the slope of the curve (convergence rate) can only be zero when hiH is 1 or O. Comparing the convergence rates for positive DA for mode (1, 2) with those for negative DA of mode (1, 2) suggests only a small change in the convergence rate. The criteria of engineering accuracy is met when the number of finite elements per side of the membrane is given by, where HIh is the reciprocal of the number of elements along a side of the square. CURVE FITTING ANALYSES We seek to enhance the estimates of Rayleigh's quotient by curve fitting the trial function, RQ(hlH) = RQ(hlH = 0) + a(hlH) = L bi(hlH)i (11) where a and bi are curve fitting constants, and, the summation is over i = 2, 3, 4 . . . , n with n selected adaptively. When hiH is small enough, Eq. (11) repre­ sents a truncated Taylor series expansion of Ray­ leigh's quotient about hiH = O. This interpreta­ tion implies that RQ is a continuous function of hiH. Under the additional assumption that con­ vergence of a sequence of Rayleigh quotients is asymptotic, we can limit the trial functions to cases for which a = 0, that is, RQ(hIH) = RQ(hIH = 0) + L bi(hIH)i. (12) In this section we will examine use of the trial functions defined by Eq. (12) in representing the relationship between accuracy of the estimates of the quotient and relative grid interval. Table 2 lists sampling of FEA grids for three different sampling strategies. Some of the quo­ tient estimates are higher than the Eq. (3) value and some less. The data of Table 2 indicates that each of the three strategies leads to Rayleigh quotients of en­ gineering accuracy for the analytical solution, Eq. (3). The conventional strategy furnishes quotients with the smallest range in the number of DA, 1.3 to 1.4. The strictly monotonic strategy produces the highest range of D while meeting the 1.3 digit requirement: 1.88 to 2.7. Table 3 defines the values of calculations in­ dex as a function of the vibration mode and the FEA sampling strategy. Table 3 lists calculation indicies for the three grid refinement strategies for NM = 1 to 10. These data indicate that index reductions using curve fitting analysis are better characterized by percentage reductions than by powers of 10, that the smallest HIh strategy tends to minimize the index (up to 62% lower than that of the conventional FEA), and that use of the strictly monotonic strategy does not mini­ mize the index. SENSITIVITY OF THE CALCULATIONS INDEX The data of Table 4 define values of the calcula­ tions index for various DA and NM. The data suggest that the index is given by, CI = 12*DA*NM2. (13) We conclude that the calculations index in­ creases linearly with the DA and quadratically with the NM required. CONCLUSIONS The analyses of the vibrating square drumhead lead to the following conclusions: 1. Finite element analyses can produce unac­ ceptable results when the grid is too coarse to represent the structural system. (Selec­ tive sampling of FEA results is a means for rendering oscillating convergence curves by curves that reflect strictly monotonic convergence.) 2. Curve fitting analysis reduces the calcula­ tions index for all cases of Table 3. Reduc­ tions were up to 52% of the index associ­ ated with conventional strategy analyses. 3. The calculations index for drumhead analy­ sis increases monotonically with increases in the NM and DA required. (The calcula­ tions index indicates that the data process­ ing resources increase linearly with the DA and NM required by the analyst.) It remains to verify which of the conclusions are valid for more general geometries than square elements and drumheads. 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Robert J. Melosh. Modeling Accuracy in FEA of Vibrations of a Drumhead, Shock and Vibration, DOI: 10.3233/SAV-1993-1103