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A PRACTICAL ERROR ESTIMATOR FOR FINITE ELEMENT PREDICTED NATURAL FREQUENCIES OF MEMBRANE VIBRATION PROBLEMS
Journal of Sound and Vibration (1996) 195(5), 739–756
A PRACTICAL ERROR ESTIMATOR FOR FINITE ELEMENT PREDICTED NATURAL FREQUENCIES OF MEMBRANE VIBRATION PROBLEMS C Z G. P. S Finite Element Analysis Research Centre, Building J07, Engineering Faculty University of Sydney, Sydney, N.S.W. 2006, Australia (Received 23 December 1994, and in final form 6 February 1996) Based on the asymptotic solution for finite element predicted natural frequencies of a membrane vibration problem, the concept of asymptotic error and a practical error estimator are presented in this paper. The present practical error estimator contains two criteria: one is the error estimator criterion and the other is the finite element mesh design criterion. By using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a membrane vibration problem be directly evaluated without any further finite element calculation, but also a new target finite element mesh for the solution of desired accuracy can be immediately designed from the relevant information of an original finite element solution. Generally, for the purpose of designing a new target finite element mesh, this original finite element solution is obtainable from a very coarse mesh of a few elements and usually does not satisfy the accuracy requirement. Since the new target finite element mesh could result in a finite element solution of desired accuracy, the finite element solution so obtained can be used for structural design in engineering practice. The related numerical results from vibration problems of five representative membranes of different shapes have demonstrated the correctness and applicability of the present practical error estimator. 7 1996 Academic Press Limited
1. INTRODUCTION
This is the second paper reporting results of a recent study associated with the development of a practical error estimator for finite element predicted natural frequencies of membrane vibration problems in engineering practice. In the first paper [1], an asymptotic formula was presented for correcting the finite element predicted natural frequencies of membrane vibration problems. Since the corrected solution obtained from the asymptotic formula is more accurate than the original finite element solution, it might be adopted as an alternative solution against which the original finite element solution could be compared. Besides, it is noted that the asymptotic solution is obtainable directly from the original finite element solution simply by using the related asymptotic formula without any further finite element calculation. Based on this consideration, a practical error estimator, which is of an asymptotic nature, is presented for finite element predicted natural frequencies of membrane vibration problems in this paper. Since the corrected solution of the finite element predicted natural frequency tends to the exact solution in the limit of finite element sizes approaching zero, the accuracy of the error estimator increases with the refinement of the finite element mesh. This fact is considered in the process of developing the present practical error estimator which contains two criteria: one is referred to as the error estimation criterion and the other is the finite element mesh design criterion. 739 0022–460X/96/350739 + 18 $18.00/0
7 1996 Academic Press Limited
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. . .
Vibrations of membranes have been considered in various papers and books for over a century [2–13]. Analytical methods have been used to determine the free vibrations of membranes with regular boundary shapes, while numerical methods have been used to deal with vibrations of membranes with complicated boundary shapes. Laura et al. [7] classified the numerical methods (see reference [7] and the references therein) into the following nine categories: (1) conformal mapping method; (2) finite difference method; (3) point matching method; (4) Rayleigh-Ritz method; (5) Galerkin method; (6) method of moments; (7) perturbation method; (8) impedance method; (9) finite element method. These methods are of different theoretical bases so that each method has a number of advantages and disadvantages. This enables some of these methods to be used successfully for solving plate and shell vibration problems [14–17]. However, Laura et al. [6] pointed out that ‘‘the finite element method is probably the most general approach at present for a very large variety of scientific and technological problems’’. Indeed, what has happened to the finite element method during the last two decades has demonstrated the correctness of this statement. Finite element solutions have been widely accepted in the design of structures for the following main reasons. (1) For most engineering problems, it is impossible to obtain exact solutions because geometry, material distribution, boundary conditions and so forth are often of a complicated nature. In addition, the material properties of a structure, although predictable and controllable, are usually within a certain permitted range as the consequence of operation and construction. (2) If the exact solution is not obtainable for a practical engineering problem, it is important and necessary to obtain an approximate solution so that a reasonable structural design can be made. In this regard, the finite element solution is an ideal approximate one, provided the error of the finite element solution is within the error tolerance of a structural design. (3) Owing to the versatility of the finite element method, it can provide approximate solutions for almost all practical engineering problems. Since the accuracy of the finite element solution is capable of being improved by mesh refinement, it is possible to obtain a finite element solution of desired accuracy. Clearly, a finite element solution is of doubtful value for a structural design unless its error is evaluated and its accuracy is within the permitted range. For this reason, error estimation of the finite element solution has become a very important topic among finite element analysts in recent years. For example, Babuska et al. [18–20], Kelly et al. [21, 22] and Oden et al. [23, 24] established the mathematical basis for error estimation of finite element solutions. Realizing the cost of computations associated with error estimation and the difficulty of implementing such computations into an existing finite element analysis code structure, Zienkiewicz and Zhu [25, 26] developed some simple but practical a posteriori error estimators for static problems in engineering practice. Although significant research [18–27] has been done on error estimation of finite element solutions of static problems, very limited research [28–31] has been carried out for the practical error estimation of finite element solutions of dynamic problems. Therefore, to meet the need for a structural design under dynamic loading, developing a practical error estimator for the finite element solution to a membrane vibration problem is the main purpose of this study. The contents of this paper are arranged as follows. In the second section, the concept of asymptotic error is presented and a practical error estimator, which contains two particular criteria, is developed for finite element predicted natural frequencies of membrane vibration problems. In the third section, this practical error estimator is applied to solve vibration problems of five representative membranes of different shapes so as to verify the correctness of the practical error estimator. Finally, some conclusions relating to this study are summarized in the fourth section.
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2. ASYMPTOTIC ERROR AND PRACTICAL ERROR ESTIMATOR
For practical engineering problems, it is impossible to obtain exact solutions so that approximate solutions are often needed for structural design. The finite element solution is one such an approximate solution that might be widely used in engineering practice if its error could be evaluated. Owing to the lack of exact solutions, it is difficult and often impossible to work out the exact error of a finite element solution for a practical engineering problem such as the vibration of a membrane of arbitrary shape. In order to circumvent this difficulty, it is necessary to evaluate the asymptotic error of a finite element solution instead of the exact error. The asymptotic error is an approximate error but tends to the exact error if the characteristic length of the elements approaches zero. Since the asymptotic solution of the finite element predicted natural frequency of a membrane vibration problem was made available recently [1], it is possible to use this solution to calculate the asymptotic error of a finite element solution for a membrane vibration problem. The asymptotic absolute error and the asymptotic relative error for the finite element predicted natural frequency of a membrane can be defined as, respectively, dia = vi − v¯ i ,
dir = (vi − v¯ i )/v¯ i ,
(1, 2)
where vi is the originally predicted natural frequency of the ith mode of a membrane in the finite element analysis and v¯ i is the corresponding asymptotic natural frequency of vi in the sense that the characteristic length of the finite elements used tends to zero so that the finite element discretized system approaches a continuum one. The expression for v¯ i was derived for membrane vibration problems in the first paper of this study [1] and is expressed as v¯ i = lim vi = z2Cki h:0
(3)
where C is the wave speed of the membrane, h is the characteristic length of finite elements used and ki is the wave number in correspondence with vi of the discretized finite element system, which can be calcuated from the equation cos (ki h) = (1 − 2bi )/(1 + bi )
(4)
where bi is a parameter expressed by the formula bi = vi2 h 2/12C 2.
(5)
For the membrane transverse vibration problems, the wave speed is defined as [2, 3] C = zT/r,
(6)
where T is the uniform tension in the membrane and r is the mass per unit area of the membrane. With the error in vi denoted as Dvi , the corresponding error in ki , can be evaluated by differentiating both sides of equation (4). This leads to the following expression for Dki : Dki = [hvi /2C 2(1 + bi )2 sin (ki h)]Dvi .
(7)
From equation (3), the error in v¯ i can be derived as Dv¯ i = z2CDki .
(8)
This error is caused by the error, Dki . Substituting equation (7) into equation (8) yields Dv¯ i = [z2hvi /2C(1 + bi )2 sin (ki h)]Dvi
(9)
. . .
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Thus, the rate of error propagation from Dvi to Dv¯ i can be expressed as fi =
Dv¯ i z2hvi z2 z2 = = = . Dvi 2C(1 + bi )2 sin (ki h) (1 + bi )z2 − bi z2 + bi (3 − bi2 )
(10)
This equation indicates that the smaller the value of fi the more accurate the value of v¯ i . In order to compare the error propagation rate of Dvi with that of Dvj (j q i), equation (4) is used to examine the range of variation of kh. If cos (kh) is limited to varying from one to zero, then kh varies from zero to p/2 and b varies from zero to 0·5. This leads to the relation 0 E h E z6C/v,
(11)
where v is the maximum circular natural frequency of interest. If a finite element mesh is comprised of elements whose characteristic length satisfies relation (11), the mesh is called as a reasonable mesh in this study. It is noted that in the finite element analysis, bi is greater than zero since the characteristic length of elements is greater than zero. As a result, fi is always less than one for a reasonable mesh. This proves the fact that, in the process of using the asymptotic formula, v¯ i is always more accurate than vi for a reasonable mesh in the finite element analysis. From the definition in equation (4), one can deduce the following equations: sin (ki h) = z1 − cos2 (ki h) = z3bi (2 − bi )/(1 + bi ), sin (kj h) = z1 − cos2 (kj h) = z3bj (2 − bj )/(1 + bj ),
(12)
By using equation (10), the following equation can be derived: fj (1 + bi )2 sin (ki h)vj (1 + bi )z2 − bi = = = fi (1 + bj )2 sin (kj h)vi (1 + bj )z2 − bj
X
2 + 3bi − bi3 . 2 + 3bj − bj3
(13)
Note that, in the case of bi Q bj Q 1, (2 + 3bj − bj3 ) − (2 + 3bi − bi3 ) = (bj − bi )[3 − (bj2 + bi bj + bi3 )] q 0.
(14)
Since the denominator is greater than the numerator in equation (13), it has been proven that fj /fi Q 1 if j q i. This indicates that the error propagation rate of a higher natural frequency is smaller than that of a lower natural frequency when equation (3) is used to obtain an asymptotic solution for a reasonable mesh in the finite element analysis. This also implies that, in the process of using the asymptotic formula, the accuracy of the asymptotic relative error for a higher natural frequency is higher than that for a lower natural frequency. That is to say, in terms of using the asymptotic formula, it is a natural phenomenon that higher natural frequencies converge quicker than lower ones. Therefore, if the accuracy of the natural frequency of mode i is of interest, the asymptotic relative error of the natural frequency of mode i + n can be used to evaluate the possible range of exact error for the natural frequency of mode i. The reason for this is that, in the finite element analysis, the exact error of a lower natural frequency is generally lower than that of a higher natural frequency. In other words, if the asymptotic relative error of the natural frequency of mode i + m (m = 1, 2, . . . , n) is within the given range of error tolerance, the real error of the natural frequency of mode i can be assured within this range. Because of the importance of n, which may be termed as a shift factor, it needs to be very carefully chosen. Generally, the value of n should be large enough so that the real error of the natural frequency of mode i can be guaranteed within a given range of error tolerance. For membrane vibration problems, numerical experience has demonstrated that the value
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of n should be equal to or greater than six for a reasonable finite element mesh and therefore, the value of n is chosen as six in this study. In order to establish a practical error estimator for the finite element solution of a membrane vibration problem, the following two basic questions must be answered. (1) For a given finite element solution, how can its accuracy be evaluated? (2) If the accuracy of a given finite element solution is not satisfactory, how can a satisfactory finite element solution, the real error of which is controlled within the desired range of error tolerance, be obtained for the purpose of a structural design in engineering practice? Since the natural frequency of a dynamic system is a very important characteristic of the system, the accuracy of the finite element predicted natural frequency of a membrane can be used to represent the overall accuracy of the finite element solution for the membrane vibration problem. Therefore, the first criterion of the present practical error estimator in relation to answering question (1) is as follows. For the ith natural frequency predicted from the finite element analysis of a membrane, an upper bound of its real relative error can be provided by the asymptotic relative error of the (i + n)th natural frequency, where n is equal to or greater than six for a reasonable mesh. To answer question two raised above, the adaptive mesh h-refinement scheme is adopted in the finite element analysis since it does not change the element type used but a satisfactory finite element solution is achieved. It has been recognized that, in finite element analysis, the general convergence characteristic of a natural frequency of a system is that, for a linear displacement element, the natural frequency converges as the square of the characteristic length of the elements used. When the characteristic length of the elements used is halved, the error in a specific natural frquency is quartered. This fact leads to the equation: N = (1/ln 4)(ln dir+ n − ln dri ), (15) where dri is the maximum relative error tolerance for the maximum natural frequency, vi of interest, dir+ n is the asymptotic relative error of the natural frequency vi + n and N is the number of times needed to repeat the h-refinement scheme, under the assumption that each existing element is divided into four elements each time. Once the value of N is obtained from equation (15), a new finite element mesh, which results in a finite element solution with accuracy in the given range of error tolerance, can be designed and generated. For example, if N is equal to three, this means that each existing element needs to be divided into 4N elements, namely 64 new elements. It should be pointed out that if N is not an integer then, from equation (15), it is necessary to let N be equal
Figure 1. Finite element model of an L-shaped cantilever membrane (C = 1 m/s).
. . .
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T 1 Asymptotic relative error for the uncorrected natural frequencies (L-shaped membrane) Frequency number
Solution (rad/s)
Mesh 1 3 elements
Mesh 2 12 elements
Mesh 3 48 elements
Mesh 4 192 elements
Mesh 5 768 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
0·186331 0·183118 1·17544 0·442241 0·406983 8·6632 0·588909 0·518206 13·6438 0·721674 0·612805 17·7655 0·866249 0·721038 20·1392
0·178849 0·178112 0·4136 0·396941 0·389225 1·9825 0·538662 0·520196 3·5498 0·718775 0·677878 6·0331 0·827418 0·765609 7·6813 0·923442 0·844769 9·3129 1·118590 0·993074 12·6392 1·169710 1·030520 13·5071 1·226170 1·071350 14·4510 1·311210 1·132070 15·8244
0·176684 0·176505 0·1015 0·385353 0·383514 0·4795 0·519036 0·514590 0·8641 0·671375 0·661899 1·4316 0·775082 0·760685 1·8927 0·830683 0·813091 2·1636 1·003920 0·973665 3·1073 1·033120 1·000310 3·2805 1·110440 1·070230 3·7572 1·191440 1·142500 4·2836
0·175988 0·175943 0·0253 0·382191 0·381738 0·1187 0·514045 0·512946 0·2143 0·659348 0·657037 0·3517 0·761615 0·758065 0·4683 0·805741 0·801544 0·5236 0·971254 0·963949 0·7578 0·990682 0·982936 0·7880 1·073580 1·063760 0·9233 1·129270 1·117870 1·0198
0·175749 0·175735 0·0079 0·381273 0·381159 0·0299 0·512786 0·512513 0·0533 0·656333 0·655760 0·0874 0·758144 0·757260 0·1167 0·799456 0·798420 0·1298 0·962765 0·961167 0·1881 0·979903 0·977998 0·1947 1·064430 1·061990 0·2296 1·113710 1·110920 0·2513
2 3 4 5 6 7 8 9 10
to the smallest integer whose value is greater than the calculated N, for the purpose of assuring the accuracy of the finite element soution from the new mesh. Thus, the second criterion of the present practical error estimator in relation to answering question two can be stated as follows. For a membrane vibration problem, a finite element solution of desired accuracy can be obtained from a new finite element mesh, which is a subdivision of elements used in the original finite element analysis and can be designed by using equation (15).
3. VERIFICATION AND APPLICATION OF THE PRACTICAL ERROR ESTIMATOR
In this section, the practical error estimator is applied to the finite element solution of five representative membrane vibration problems. To reflect the applicability of the present practical error estimator to various problems of different geometric shapes, the domains of these problems are selected as follows: (1) an L-shaped cantilever membrane; (2) a trapezoidal cantilever membrane; (3) a tapered cantilever membrane with a circular hole in the centre; (4) a skewed cantilever membrane; (5) a circular hollow cantilever membrane. For the purpose of verifying the correctness of the practical error estimator and providing a series of finite element solutions for these representative membrane vibration problems,
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the adaptive mesh h-refinement scheme, which results in a monotonic convergence consequence of the finite element solution, is adopted herein. Conventional four-node finite elements are used for the first four problems and eight-node isoparametric elements are used for the fifth problem in the finite element analysis. The wave speed of the membrane is equal to 1 m/s throughout this study. 3.1. - As shown in Figure 1, an L-shaped cantilever membrane is initially divided into three square finite elements of equal areas. In this case, the characteristic length of the elements is equal to the side length of the elements used. Based on the adaptive mesh h-refinement scheme, the domain of the L-shaped cantilever membrane is further subdivided into 12, 48, 192 and 768 square elements. Since all finite elements used in the process of discretization are of equal areas, the characteristic lengths of the elements are 5, 2·5, 1·25, 0·625 and 0·3125 m, respectively, when the L-shaped membrane is divided into 3, 12, 48, 92 and 768 elements. Table 1 shows the asymptotic relative error for the originally predicted natural frequencies of the L-shaped cantilever membrane from the finite element analysis. In this table, the ‘‘uncorrected solution’’ is the finite element solution directly obtained from the finite element analysis, whereas the ‘‘corrected solution’’ is the asymptotic solution which is obtained by substituting the finite element solution into the asymptotic formula expressed in the last section. It can be seen from these results that, for any given natural frequency, the asymptotic relative error decreases with the refinement of a finite element mesh. For example, the asymptotic relative error of the fundamental natural frequency is 1·7544% when three elements are used to model the L-shaped membrane, but it is reduced to 0·4136%, 0·1015%, 0·0253% and 0·0079% when 12, 48, 192 and 768 elements are used. Also, for any given mesh, the asymptotic relative error monotonically increases with the increase of the natural frequency number. For mesh 1 (3 elements) as an example, the asymptotic relative error of the predicted natural frequency increases from 1·7544% to 20·1392% when the natural frequency number increases from 1 to 5. This demonstrates that the asymptotic relative error of the highest natural frequency of interest is a controlling factor in the error estimation of a finite element solution for a membrane vibration problem. In order to verify the correctness of the present practical error estimator, the finite element solution obtained from mesh 5 of 768 elements is viewed as the ‘‘exact solution’’ since the true exact solution is not available for the vibration problem of an L-shaped
Figure 2. Finite element model of a trapezoidal cantilever membrane (C = 1 m/s).
. . .
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T 2 Asymptotic relative error for the uncorrected natural frequencies (trapezoidal membrane) Frequency Solution number (rad/s) 1 2 3 4 5 6 7 8 9 10
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
Mesh 5 1024 elements
0·179418 0·177237 1·2312 0·528056 0·483723 9·1651 0·597811 0·537522 11·2160 0·838263 0·711116 17·8800 1·030620 0·858999 19·9792
0·176242 0·175712 0·3018 0·479703 0·469540 2·1645 0·546177 0·531246 2·7757 0·703367 0·673312 4·4638 0·906972 0·847159 7·0604 0·969133 0·897995 7·9219 1·023090 0·941303 8·6887 1·223360 1·095930 11·6275 1·294760 1·149010 12·6853 1·356990 1·1194540 13·5995
0·175376 0·175245 0·0749 0·461881 0·459509 0·5163 0·531306 0·527711 0·6812 0·671175 0·664006 1·0796 0·820996 0·808060 1·6008 0·889689 0·873348 1·8710 1·938925 0·919826 2·0763 1·118700 1·087120 2·9044 1·162640 1·127410 3·1247 1·220640 1·180210 3·4255
0·175154 0·175121 0·0190 0·457230 0·456648 0·1274 0·527627 0·526734 0·1695 0·662458 0·660696 0·2667 0·797502 0·794438 0·3856 0·867375 0·863441 0·4556 0·917829 0·913174 0·5097 1·083700 1·076070 0·7081 1·103730 1·095680 0·7343 1·163520 1·154120 1·8148
0·175098 0·175090 0·0048 0·456056 0·455912 0·0315 0·526709 0·526486 0·0424 0·660221 0·659784 0·0663 0·791675 0·790919 0·0956 0·861655 0·860682 0·1130 0·912661 0·911505 0·1269 1·072520 1·070650 0·1750 1·090520 1·088550 0·1809 1·149490 1·147190 0·2009
cantilever membrane. By using this ‘‘exact solution’’, the real relative error of the finite element predicted natural frequency of the membrane can be calculated from the equation dirr = (vi − viexact )/viexact ,
(16)
where dirr is the ‘‘real’’ relative error of the finite element predicted natural frequency of mode i, vi is the uncorrected natural frequency of mode i and viexact is the ‘‘exact solution’’ for the natural frequency of mode i. If only the accuracy of the fundamental natural frequency is of interest, its ‘‘real’’ relative error, d1rr , can be worked out by using equation (16), and is 6·0211% (namely, d1rr = (v1 − v1exact )/v1exact = (0·186331 − 0·175749)/0·175749 = 6·0211%), 1·7639% (namely, (0·178849 − 0·175749)/0·175749 = 1·7639%), 0·5320% (namely, (0·176684 − 0·175749)/ 0·175749 = 0·5320%) and 0·1360%(0·175988 − 0·175749)/0·175749 = 0·1360%) for the finite element meshes of 3, 12, 48 and 192 elements respectively, whereas the asymptotic relative errors d7r , are 20·1392%, 12·6392%, 3·1073% and 0·7578% for the corresponding meshes. It is noted that the value of d5r is used instead of d7r for mesh 1 because d7r is not available in this case. Since the asymptotic relative error, d7r , is greater than the ‘‘real’’ relative error, d1rr , it is concluded that the ‘‘real’’ relative error of the predicted natural
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frequency, vi , can be estimated by calculating the asymptotic relative error of the uncorrected natural frequency, vi + n , where n is equal to six for membrane vibration problems. This verifies the correctness of the first criterion of the present practical error estimator. Next, the correctness of the second criterion can be investigated by considering the finite element solution obtained from mesh 2. If the fourth natural frequency of the L-shaped membrane is of interest and its maximum ‘‘real’’ relative error tolerance, dr4 , is equal to 1%, the number of times needed to repeat the h-refinement scheme, under the assumption of each element being divided into four elements each time, can be calculated from equation (15) as r N = (1/ln 4)(ln d10 − ln dr4 ) = (1/ln 4)(ln 0·158244 − ln 0·01) = 1·9920.
(17)
Since N is not an integer, N is set to be 2, which is the smallest integer not only less than N. This requires that in order to obtain a finite element solution in which the ‘‘real’’ relative error of the fourth natural frequency is guaranteed with 1%, each existing element in mesh 2 be divided into 16 new elements. This subdivision leads to a target mesh of 192 elements. That is to say, based only on the information obtained from mesh 2 of 12 elements, a new target mesh can be designed to result in a finite element solution of desired accuracy. It is clear that the ‘‘real’’ relative error of the fourth natural frequency obtained from the mesh of 192 elements is 0·4594%, which is indeed smaller than the given maximum ‘‘real’’ relative error tolerance, dr4 = 1%. Therefore, the correctness of the second criterion of the present practical error estimator has been verified by this example. 3.2. In order to examine the correctness of the present practical error estimator when distorted elements are used in the finite element analysis, a trapezoidal cantilever membrane is selected and initially divided into four quadrilateral elements. Figure 2 shows the initial discretization of the trapezoidal cantilever membrane. Since distorted elements are used to model this problem, the equivalent characteristic length of elements needs to be calculated from the formula h = zA/NE ,
(18)
where h is the equivalent characteristic length of the elements used, A is the total area of
Figure 3. Finite element model of a trapered cantilever membrane with a circular hole in the centre (R = 1·5) C = 1 m/s).
. . .
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T 3 Asymptotic relative error for the uncorrected natural frequencies (trapered membrane with a hole) Frequency number
Solution (rad/s)
Mesh 1 8 elements
Mesh 2 32 elements
Mesh 3 128 elements
Mesh 4 512 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%) v11 v¯ 11 r d11 (%)
0·177568 0·176554 1·5739 0·474199 0·456569 3·8614 0·510789 0·489105 4·4332 0·764263 0·700806 9·0549 0·909498 0·812128 11·9894 1·174960 1·003340 17·1046 1·364710 1·140170 19·6944 1·449630 1·206650 20·1366
0·173988 0·173748 0·1381 0·456207 0·451975 0·9363 0·476962 0·473137 1·0219 0·701946 0·687042 2·1694 0·824138 0·800529 2·9491 1·004690 0·963472 4·2780 1·028270 0·984314 4·4657 1·171790 1·108920 5·6690 1·300520 1·217410 6·8272 1·341860 1·251590 7·2125 1·418480 1·314120 7·9412
0·172678 0·172619 0·0343 0·449213 0·448185 0·2294 0·464391 0·463255 0·2451 0·677495 0·673995 0·5192 0·785463 0·780035 0·6958 0·922868 0·914127 0·9562 0·939806 0·930583 0·9911 1·088690 1·074480 1·3223 1·171910 1·154290 1·5268 1·226840 1·206700 1·6691 1·249570 1·228320 1·7297
0·172230 0·172215 0·0087 0·447113 0·446858 0·0572 0·460839 0·460561 0·0603 0·669899 0·669045 0·1277 0·773575 0·772260 0·1702 0·898832 0·896774 0·2295 0·914096 0·911932 0·2373 1·053260 1·049960 0·3147 1·141140 1·136940 0·3691 1·170160 1·165640 0·3880 1·215330 1·210270 0·4183
2 3 4 5 6 7 8 9 10 11
the problem domain and NE is the total number of elements used in the finite element discretization of the problem domain. When the trapezoidal cantilever membrane shown in Figure 2 is divided into 4, 16, 64, 256 and 1024 quadrilateral elements, the equivalent characteristic lengths of the elements are 4·3301, 2·1651, 1·0825, 0·5413 and 0·2706 m, respectively. As was mentioned before, any refinement of a mesh reduces the equivalent characteristic length of the elements and therefore improves the accuracy of the finite element solution. Table 2 shows the asymptotic relative error for the originally predicted natural frequencies of the trapezoidal cantilever membrane when five different finite element meshes are used in the finite element analysis. With the refinement of the mesh, the finite element solution for any natural frequency of the trapezoidal membrane converges to the exact solution from above because the consistent mass matrix of the element was used in the analysis. The general trend of the asymptotic relative error variation with the increase of the natural frequency number and the refinement of the element mesh is the same as
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observed for the vibration of an L-shaped cantilever membrane. Besides, it can be concluded from these results that the asymptotic relative error of the highest natural frequency of interest is a controlling factor in the error estimation of a finite element solution for a membrane vibration problem, even if distorted elements are used in the finite element analysis. From the finite element solution for the trapezoidal cantilever membrane vibration problem, the validity of the present practical error estimator can be demonstrated as follows. With the second natural frequency of the trapezoidal membrane taken as an example, the ‘‘real’’ relative error of this natural frequency can be worked out by using equation (16) and is 15·7875%, 5·1815%, 1·2773% and 0·2574% for the finite element mesh of 4, 16, 64 and 256 elements, respectively, whereas the asymptotic relative errors d8r are 19·9792%, 11·6275%, 2·9044% and 0·7081% for the respective meshes. Note that the value of d5r is used instead of d8r for mesh 1 because d8r is not available in this case. Since the asymptotic relative error, d8r , is greater than the ‘‘real’’ relative error, d2rr , the validity of the first criterion of the present practical error estimator has been demonstrated, even though quadrilateral elements have been used. To examine the validity of the second criterion of the present practical estimator, it is assumed that the third natural frequency
T 4 Asymptotic relative error for the uncorrected natural frequencies (skewed membrane) Frequency Solution number (rad/s) 1 2 3 4 5 6 7 8 9 10
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
Mesh 5 1024 elements
0·157911 0·155930 1·2704 0·351378 0·332135 5·7967 0·517654 0·465458 11·2139 0·628048 0·546404 14·9421 0·689156 0·584071 16·6333 0·785241 0·658443 19·2572
0·154583 0·154106 0·3095 0·332644 0·328030 1·4066 0·517428 0·500943 3·2908 0·544078 0·525085 3·6171 0·716437 0·675897 5·9980 0·768633 0·719716 6·7967 0·892127 0·820045 8·7900 1·001740 0·905407 10·6397 1·012170 0·913353 10·8191 1·092810 0·973996 12·1986
0·153589 0·153471 0·0769 0·327157 0·326028 0·3463 0·501238 0·497227 0·8067 0·517937 0·512543 0·8573 0·675490 0·665844 1·4487 0·719136 0·707556 1·6366 0·829235 0·811731 2·1564 0·907110 0·884452 2·5618 0·923205 0·899377 2·6494 1·018780 0·987238 3·1950
0·153300 0·153270 0·0196 0·325759 0·325478 0·0863 0·496563 0·495572 0·2000 0·510233 0·509157 0·2113 0·663556 0·661200 0·3563 0·705261 0·702437 0·4020 0·807122 0·802903 0·5255 0·877943 0·872528 0·6206 0·895983 0·890231 0·6461 0·980426 0·972915 0·7720
0·153213 0·153205 0·0052 0·325408 0·325337 0·0218 0·495321 0·495074 0·0499 0·508562 0·508296 0·0523 0·660516 0·659931 0·0886 0·701721 0·701019 0·1001 0·801477 0·800432 0·1306 0·870511 0·869174 0·1538 0·888922 0·887498 0·1538 0·970677 0·968826 0·1911
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. . .
Figure 4. Finite element model of a skewed cantilever membrane (C = 1 m/s).
of the trapezoidal membrane is of interest and that its maximum ‘‘real’’ relative error tolerance, dr3 , is equal to 1%. By using equation (15) and the information from mesh 2, the value of N is worked out as 1·8325. Therefore, the value of N is taken as 2 and each element in mesh 2 should be divided into 16 new elements, which results in a new target mesh of 256 elements. The ‘‘real’’ relative error of the third natural frequency obtained from mesh 4 of 256 elements is 0·1743%, which is smaller than the given maximum ‘‘real’’ relative error tolerance, dr3 = 1%. This demonstrates that, in spite of using distorted elements in the finite element analysis, the second criterion of the present practical error estimator is also valid for the error estimation of the finite element solution of membrane vibration problems. 3.3. This is a challenging problem for checking the correctness and applicability of the present practical error estimator, since highly distorted elements must be used to model the medium around the circular hole of the membrane. As shown in Figure 3, a tapered cantilever membrane with a circular hole in the centre is initially divided into eight quadrilateral elements. In the adaptive mesh h-refinement scheme, the domain of the
Figure 5. Finite element model of a circular hollow membrane (C = 1 m/s).
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tapered cantilever membrane is further subdivided into 32, 128 and 512 quadrilateral elements. The equivalent characteristic lengths of the elements are 2·9703, 1·4813, 0·7402 and 0·3700 m for the finite element meshes of 8, 32, 128 and 512 elements, respectively. Table 3 shows the asymptotic relative error for the originally predicted natural frequencies of the tapered cantilever membrane with a central circular hole when four different finite element meshes are used in the finite element anlaysis. The finite element solution from mesh 4 of 512 elements is viewed as the ‘‘exact solution’’ since the asymptotic relative error for the fundamental natural frequency is only 0·0087% and the maximum asymptotic relative error for the first 11 natural frequencies is less than 0·5%. The same conclusions as described in sections 3.1 and 3.2 can be drawn from these numerical results shown in Table 3. With the second natural frequency of the tapered membrane taken as an example, the ‘‘real’’ relative error of this natural frequency, d2rr , can be worked out by using equation (16) and is 6·0580%, 2·0340% and 0·4697% for the finite element meshes of 8, 32 and 128 elements, respectively, whereas the respective asymptotic relative errors d8r are 20·1366%, 5·6690% and 1·3223%. This further confirms the validity of the first criterion of the present practical error estimator. If the maximum natural frequency of interest is the fifth natural frequency of the tapered membrane and its maximum ‘‘real’’ relative error tolerance, dr5 , is equal to 2%, the value of N is worked out as 0·9947 by means T 5 Asymptotic relative error for the uncorrected natural frequencies (circular hollow membrane) Frequency number
Solution (rad/s)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
0·172966 0·171236 1·0103 0·351115 0·337769 3·9512 0·503138 0·467743 7·5672 0·741935 0·650229 14·1036 0·775812 0·674558 15·0104 1·079370 0·898625 20·1135 1·141560 0·958408 19·1100
0·169294 0·168880 0·2451 0·345006 0·341573 1·0051 0·491546 0·481885 2·0048 0·676516 0·652473 3·6849 0·737379 0·706800 4·3264 0·923588 0·867132 6·5107 0·969824 0·905563 7·0962 1·109940 1·018790 8·9469 1·143740 1·045420 9·4048 1·224140 1·107690 10·5129
0·167592 0·167491 0·0603 0·342173 0·341317 0·2508 0·486853 0·484408 0·5047 0·667303 0·661082 0·9410 0·730811 0·722681 1·1250 0·908295 0·892936 1·7201 0·947490 0·930127 1·8667 1·104970 1·077910 2·5104 1·127470 1·098800 2·6092 1·160670 1·129520 2·7578
0·166803 0·166779 0·0144 0·340922 0·340711 0·0619 0·484749 0·484139 0·1260 0·664877 0·663309 0·2364 0·729409 0·727341 0·2805 0·906514 0·902562 0·4379 0·944246 0·939784 0·4748 1·104570 1·097460 0·6479 1·125130 1·117620 0·6720 1·153760 1·145670 0·7061
2 3 4 5 6 7 8 9 10
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. . .
Figure 6. Vibration modes of the L-shaped cantilever membrane.
of equation (15) and the information from mesh 2. Therefore, the value of N is taken as 1 and each element in mesh 2 should be divided into 4 new elements, which leads to a new target mesh of 128 elements. Clearly, the ‘‘real’’ relative error of the fifth natural frequency obtained from the mesh of 128 elements is 1·5368%, which is smaller than the given maximum ‘‘real’’ relative error tolerance, dr5 = 2%. This demonstrates that, although highly distorted elements have been used in the finite element analysis, the present practical error estimator is a valid error estimation of a finite element solution for the natural frequencies of membrane vibration problems. 3.4. This problem is selected to make a further investigation into the correctness and applicability of the present practical error estimator. As shown in Figure 4, a skewed cantilever membrane is initiallly discretized into 4 quadrilateral elements. In the adaptive mesh h-refinement scheme, the domain of the skewed cantilever membrane is further subdivided into 16, 64, 256 and 1024 quadrilateral elements. The equivalent characteristic lengths of the elements used are 5, 2·5, 1·25, 0·625 and 0·3125 m for the finite element meshes of 4, 16, 64, 256 and 1024 elements, respectively. Table 4 shows the asymptotic relative errors for the originally predicted natural frequencies of the skewed cantilever membrane with five different meshes. For this problem, the finite element solution from mesh 5 of 1024 elements is taken as the ‘‘exact solution’’ since the asymptotic relative error for the first fundamental natural frequency is only 0·0052% and the maximum asymptotic error for the first 10 natural frequencies
Figure 7. Vibration modes of the trapezoidal cantilever membrane.
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Figure 8. Vibration modes of the trapered cantilever membrane with L circular hole in the centre (R = 1·5).
is less than 0·2%. By following the same procedures as used in the first three examples, the validity of both the first and the second criterion of the present practical error estimator is confirmed by the numerical results shown in Table 4. This demonstrates that the present error estimator is applicable to membranes of various geometries. 3.5. This is a curvilinear membrane vibration problem. For this problem, eight-node isoparametric elements are used to model the curved geometry of the membrane. As shown in Figure 5, a circular hollow cantilever membrane is initially divided into four eight-node isoparametric elements. By using the adaptive mesh h-refinement scheme, the circular hollow membrane is further subdivided into 16, 64 and 256 eight-node isoparametric elements. The equivalent characteristic lengths of the elements used are 4·0612, 2·0306, 1·0153 and 0·5077 m for the finite element meshes of 4, 16, 64 and 256 elements respectively. Table 5 shows the asymptotic relative error for the originally predicted natural frequencies of the circular hollow cantilever membrane when four different meshes are used in the finite element analysis. In this case, the finite element solution from mesh 4 of 256 eight-node isoparametric elements is viewed as the ‘‘exact solution’’ because the asymptotic relative error for the fundamental natural frequency is only 0·0144% and the maximum asymptotic relative error for the first 10 natural frequencies is less than 0·8%. The validity
Figure 9. Vibration modes of the skewed cantilever membrane.
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. . .
Figure 10. Vibration modes of the circular hollow membrane.
of the present practical error estimator can be verified from the numerical results shown in Table 5, obtained by the same procedures as before. This demonstrates that the present error estimator can be applied to curvilinear membrane vibration problems. Since the aforementioned five representative problems have been studied in great detail, they may serve as benchmark problems, against which any new numerical solution could be compared. For this reason and for the sake of completeness, the first six vibration modes of the L-shaped cantilever membrane, the trapezoidal cantilever membrane, the tapered membrane with a central circular hole, the skewed cantilever membrane and the circular hollow cantilever membrane are shown in Figures 6–10 respectively. All the vibration modes are exaggerated in these figures because, otherwise they are too small to be seen. It can be seen from these figures that the vibration mode of a membrane depends not only on the natural frequency number, but also on the particular shape of the membrane. Generally speaking, for a specific membrane, the larger the natural frequency number, the more complicated the shape of the vibration mode.
4. CONCLUSIONS
The concept of asymptotic error and a practical error estimator for predicted natural frequencies of a membrane vibration problem in the finite element analysis have been presented. The practical error estimator contains the following two criteria: (1) For the ith natural frequency predicted from the finite element analysis, an upper bound of its real relative error can be provided by the asymptotic relative error of the (i + n)th natural frequency, where n is equal to or greater than six for a reasonable mesh of a membrane; (2) for a membrane vibration problem, a finite element solution of desired accuracy can be obtained from a new finite element mesh, which is a subdivision of elements used in the original finite element analysis and can be designed by using equation (15). By using the first criterion, the accuracy of a finite element solution for the natural frequencies of a membrane vibration problem can be evaluated, and by using the second criterion, a target finite element mesh which produces a finite element solution of desired accuracy, can be designed. The correctness and applicability of the present practical error estimator have been demonstrated by the numerical results from vibration problems of five representative membranes of different shapes.
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ACKNOWLEDGMENTS
The authors are grateful to the referees for their valuable suggestions and comments on an earlier draft of this paper.
REFERENCES 1. C. Z and G. P. S 1996 Communications in Numerical Methods in Engineering 12, 63–73. An asymptotic formula for correcting finite element predicted natural frequencies of membrane vibration problems. 2. L R 1877 Theory of Sound. New York: Dover Publications. 3. P. M. M 1936 Vibration and Sound. New York: McGraw-Hill Book Co. 4. G. R. S 1967 Journal of Sound and Vibration 5, 1–8. Finite transform solution of the symmetrically vibrating annular membrane. 5. N. Y. O 1967 Journal of Sound and Vibration 6, 365–374. General solution to the equation of the vibrating membrane. 6. P. A. A. L 1967 Journal of the Acoustical Society of America 42, 21–26. Calculation of eigenvalues for uniform fluid waveguides with complicated cross sections. 7. P. A. A. L, E. R and M. J. M 1972 Journal of Sound and Vibration 20, 27–38. On the analysis of waveguides of doubly-connected cross-section by the method of conformal mapping. 8. K. N 1978 Journal of Applied Mechanics 45, 153–158. Vibrations and dynamic response of membranes with arbitrary shape. 9. K. S 1976 Journal of Sound and Vibration 19, 535–540. Free vibration analysis of a composite rectangular membrane consisting of strips. 10. J. P. S and C. O. H 1983 Journal of Sound and Vibration 87, 71–81. Bounds on natural frequencies of composite circular membranes: Integral equation methods. 11. P. A. A. L, L. E and R. O. G 1985 Journal of Sound and Vibration 101, 299–306. Transverse vibration of composite membranes of arbitrary boundary shape. 12. H. P. W. G 1986 Journal of Sound and Vibration 110, 395–411. New types of vibration modes for stepped membranes. 13. V. H. C and P. A. A. L 1992 Journal of Sound and Vibration 156, 217–225. Vibrations of non-homogeneous rectangular membranes. 14. A. W. L 1969 Vibration of Plates (NASA SP-160). Washington DC: U.S. Government Printing Office. 15. A. W. L 1973 Vibration of Plates (NASA SP-288). Washington DC: U.S. Government Printing Office. 16. A. W. L 1967 Journal of Sound and Vibration 6, 145–148. Vibration of a simply supported elliptical plate. 17. A. W. L, J. C. MB and R. E. K 1984 Journal of Sound and Vibration 96, 159–173. Vibration of twisted cantilevered plates—summary of previous and current studies. 18. I. B and W. C. R 1978 SIAM Journal of Numerical Analysis 15, 736–754. Error estimates for adaptive finite element computation. 19. I. B and W. C. R 1978 International Journal of Numerical Methods in Engineering 12, 1597–1615. A posteriori error estimates for the finite element method. 20. I. B and A. M 1987 Computer Methods in Applied Mechanical Engineering 61, 1–40. A feedback finite element method with a posteriori error estimation: Part I—The finite element method and some basic properties of the a posteriori error estimator. 21. D. W. K 1984 International Journal of Numerical Methods in Engineering 20, 1491–1506. The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method. 22. D. W. K, J. P. D, S. R. G and O. C. Z 1983 International Journal of Numerical Methods in Engineering 19, 1593–1619. A posteriori error analysis and adaptive processes in the finite element method: Part 1—Error analysis. 23. J. T. O, L. D, W. R and T. A. W 1989 Computer Methods in Applied Mechanical Engineering 77, 113–180. Towards a univeral h-p adaptive finite element strategy: Part 2—A-posteriori error estimation.
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24. W. R, J. T. O and L. D 1989 Computer methods in Applied Mechanical Engineering 77, 181–212. Toward a universal h-p adaptive finite element strategy: Part 3—Design of h-p meshes. 25. O. C. Z and J. Z. Z 1987 International Journal of Numerical Methods in Engineering 24, 337–357. A simple estimator and adaptive procedure for practical engineering analysis. 26. O. C. Z and J. Z. Z 1992 International Journal of Numerical Methods in Engineering 33, 1365–1382. The superconvergent patch recovery and a posteriori error estimates: Part 2—Error estimates and adaptivity. 27. C. Z and G. P. S 1995 Communications in Numerical Methods in Engineering 11, 13–23. Analytical solutions of mass transport problems for error estimation of finite/infinite element methods. 28. R. D. C and J. A 1992 Computer and Structures 44, 619–626. Error estimation and adaptive meshing for vibration problems. 29. Y. M. X and G. P. S 1993 Communications in Numerical Methods in Engineering 9, 671–680. Explicit formulas for correcting finite element predictions of natural frequencies. 30. C. Z and G. P. S 1996 Computers and Structures (in press). A posteriori error estimators/correctors for natural frequencies of membrane vibration problems. 31. C. Z and G. P. S 1996 Computers and Structures 59, 949–963. A posteriori error estimators/correctors for natural frequencies of thin plate vibration problems.
A PRACTICAL ERROR ESTIMATOR FOR FINITE ELEMENT PREDICTED NATURAL FREQUENCIES OF MEMBRANE VIBRATION PROBLEMS C Z G. P. S Finite Element Analysis Research Centre, Building J07, Engineering Faculty University of Sydney, Sydney, N.S.W. 2006, Australia (Received 23 December 1994, and in final form 6 February 1996) Based on the asymptotic solution for finite element predicted natural frequencies of a membrane vibration problem, the concept of asymptotic error and a practical error estimator are presented in this paper. The present practical error estimator contains two criteria: one is the error estimator criterion and the other is the finite element mesh design criterion. By using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a membrane vibration problem be directly evaluated without any further finite element calculation, but also a new target finite element mesh for the solution of desired accuracy can be immediately designed from the relevant information of an original finite element solution. Generally, for the purpose of designing a new target finite element mesh, this original finite element solution is obtainable from a very coarse mesh of a few elements and usually does not satisfy the accuracy requirement. Since the new target finite element mesh could result in a finite element solution of desired accuracy, the finite element solution so obtained can be used for structural design in engineering practice. The related numerical results from vibration problems of five representative membranes of different shapes have demonstrated the correctness and applicability of the present practical error estimator. 7 1996 Academic Press Limited
1. INTRODUCTION
This is the second paper reporting results of a recent study associated with the development of a practical error estimator for finite element predicted natural frequencies of membrane vibration problems in engineering practice. In the first paper [1], an asymptotic formula was presented for correcting the finite element predicted natural frequencies of membrane vibration problems. Since the corrected solution obtained from the asymptotic formula is more accurate than the original finite element solution, it might be adopted as an alternative solution against which the original finite element solution could be compared. Besides, it is noted that the asymptotic solution is obtainable directly from the original finite element solution simply by using the related asymptotic formula without any further finite element calculation. Based on this consideration, a practical error estimator, which is of an asymptotic nature, is presented for finite element predicted natural frequencies of membrane vibration problems in this paper. Since the corrected solution of the finite element predicted natural frequency tends to the exact solution in the limit of finite element sizes approaching zero, the accuracy of the error estimator increases with the refinement of the finite element mesh. This fact is considered in the process of developing the present practical error estimator which contains two criteria: one is referred to as the error estimation criterion and the other is the finite element mesh design criterion. 739 0022–460X/96/350739 + 18 $18.00/0
7 1996 Academic Press Limited
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. . .
Vibrations of membranes have been considered in various papers and books for over a century [2–13]. Analytical methods have been used to determine the free vibrations of membranes with regular boundary shapes, while numerical methods have been used to deal with vibrations of membranes with complicated boundary shapes. Laura et al. [7] classified the numerical methods (see reference [7] and the references therein) into the following nine categories: (1) conformal mapping method; (2) finite difference method; (3) point matching method; (4) Rayleigh-Ritz method; (5) Galerkin method; (6) method of moments; (7) perturbation method; (8) impedance method; (9) finite element method. These methods are of different theoretical bases so that each method has a number of advantages and disadvantages. This enables some of these methods to be used successfully for solving plate and shell vibration problems [14–17]. However, Laura et al. [6] pointed out that ‘‘the finite element method is probably the most general approach at present for a very large variety of scientific and technological problems’’. Indeed, what has happened to the finite element method during the last two decades has demonstrated the correctness of this statement. Finite element solutions have been widely accepted in the design of structures for the following main reasons. (1) For most engineering problems, it is impossible to obtain exact solutions because geometry, material distribution, boundary conditions and so forth are often of a complicated nature. In addition, the material properties of a structure, although predictable and controllable, are usually within a certain permitted range as the consequence of operation and construction. (2) If the exact solution is not obtainable for a practical engineering problem, it is important and necessary to obtain an approximate solution so that a reasonable structural design can be made. In this regard, the finite element solution is an ideal approximate one, provided the error of the finite element solution is within the error tolerance of a structural design. (3) Owing to the versatility of the finite element method, it can provide approximate solutions for almost all practical engineering problems. Since the accuracy of the finite element solution is capable of being improved by mesh refinement, it is possible to obtain a finite element solution of desired accuracy. Clearly, a finite element solution is of doubtful value for a structural design unless its error is evaluated and its accuracy is within the permitted range. For this reason, error estimation of the finite element solution has become a very important topic among finite element analysts in recent years. For example, Babuska et al. [18–20], Kelly et al. [21, 22] and Oden et al. [23, 24] established the mathematical basis for error estimation of finite element solutions. Realizing the cost of computations associated with error estimation and the difficulty of implementing such computations into an existing finite element analysis code structure, Zienkiewicz and Zhu [25, 26] developed some simple but practical a posteriori error estimators for static problems in engineering practice. Although significant research [18–27] has been done on error estimation of finite element solutions of static problems, very limited research [28–31] has been carried out for the practical error estimation of finite element solutions of dynamic problems. Therefore, to meet the need for a structural design under dynamic loading, developing a practical error estimator for the finite element solution to a membrane vibration problem is the main purpose of this study. The contents of this paper are arranged as follows. In the second section, the concept of asymptotic error is presented and a practical error estimator, which contains two particular criteria, is developed for finite element predicted natural frequencies of membrane vibration problems. In the third section, this practical error estimator is applied to solve vibration problems of five representative membranes of different shapes so as to verify the correctness of the practical error estimator. Finally, some conclusions relating to this study are summarized in the fourth section.
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2. ASYMPTOTIC ERROR AND PRACTICAL ERROR ESTIMATOR
For practical engineering problems, it is impossible to obtain exact solutions so that approximate solutions are often needed for structural design. The finite element solution is one such an approximate solution that might be widely used in engineering practice if its error could be evaluated. Owing to the lack of exact solutions, it is difficult and often impossible to work out the exact error of a finite element solution for a practical engineering problem such as the vibration of a membrane of arbitrary shape. In order to circumvent this difficulty, it is necessary to evaluate the asymptotic error of a finite element solution instead of the exact error. The asymptotic error is an approximate error but tends to the exact error if the characteristic length of the elements approaches zero. Since the asymptotic solution of the finite element predicted natural frequency of a membrane vibration problem was made available recently [1], it is possible to use this solution to calculate the asymptotic error of a finite element solution for a membrane vibration problem. The asymptotic absolute error and the asymptotic relative error for the finite element predicted natural frequency of a membrane can be defined as, respectively, dia = vi − v¯ i ,
dir = (vi − v¯ i )/v¯ i ,
(1, 2)
where vi is the originally predicted natural frequency of the ith mode of a membrane in the finite element analysis and v¯ i is the corresponding asymptotic natural frequency of vi in the sense that the characteristic length of the finite elements used tends to zero so that the finite element discretized system approaches a continuum one. The expression for v¯ i was derived for membrane vibration problems in the first paper of this study [1] and is expressed as v¯ i = lim vi = z2Cki h:0
(3)
where C is the wave speed of the membrane, h is the characteristic length of finite elements used and ki is the wave number in correspondence with vi of the discretized finite element system, which can be calcuated from the equation cos (ki h) = (1 − 2bi )/(1 + bi )
(4)
where bi is a parameter expressed by the formula bi = vi2 h 2/12C 2.
(5)
For the membrane transverse vibration problems, the wave speed is defined as [2, 3] C = zT/r,
(6)
where T is the uniform tension in the membrane and r is the mass per unit area of the membrane. With the error in vi denoted as Dvi , the corresponding error in ki , can be evaluated by differentiating both sides of equation (4). This leads to the following expression for Dki : Dki = [hvi /2C 2(1 + bi )2 sin (ki h)]Dvi .
(7)
From equation (3), the error in v¯ i can be derived as Dv¯ i = z2CDki .
(8)
This error is caused by the error, Dki . Substituting equation (7) into equation (8) yields Dv¯ i = [z2hvi /2C(1 + bi )2 sin (ki h)]Dvi
(9)
. . .
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Thus, the rate of error propagation from Dvi to Dv¯ i can be expressed as fi =
Dv¯ i z2hvi z2 z2 = = = . Dvi 2C(1 + bi )2 sin (ki h) (1 + bi )z2 − bi z2 + bi (3 − bi2 )
(10)
This equation indicates that the smaller the value of fi the more accurate the value of v¯ i . In order to compare the error propagation rate of Dvi with that of Dvj (j q i), equation (4) is used to examine the range of variation of kh. If cos (kh) is limited to varying from one to zero, then kh varies from zero to p/2 and b varies from zero to 0·5. This leads to the relation 0 E h E z6C/v,
(11)
where v is the maximum circular natural frequency of interest. If a finite element mesh is comprised of elements whose characteristic length satisfies relation (11), the mesh is called as a reasonable mesh in this study. It is noted that in the finite element analysis, bi is greater than zero since the characteristic length of elements is greater than zero. As a result, fi is always less than one for a reasonable mesh. This proves the fact that, in the process of using the asymptotic formula, v¯ i is always more accurate than vi for a reasonable mesh in the finite element analysis. From the definition in equation (4), one can deduce the following equations: sin (ki h) = z1 − cos2 (ki h) = z3bi (2 − bi )/(1 + bi ), sin (kj h) = z1 − cos2 (kj h) = z3bj (2 − bj )/(1 + bj ),
(12)
By using equation (10), the following equation can be derived: fj (1 + bi )2 sin (ki h)vj (1 + bi )z2 − bi = = = fi (1 + bj )2 sin (kj h)vi (1 + bj )z2 − bj
X
2 + 3bi − bi3 . 2 + 3bj − bj3
(13)
Note that, in the case of bi Q bj Q 1, (2 + 3bj − bj3 ) − (2 + 3bi − bi3 ) = (bj − bi )[3 − (bj2 + bi bj + bi3 )] q 0.
(14)
Since the denominator is greater than the numerator in equation (13), it has been proven that fj /fi Q 1 if j q i. This indicates that the error propagation rate of a higher natural frequency is smaller than that of a lower natural frequency when equation (3) is used to obtain an asymptotic solution for a reasonable mesh in the finite element analysis. This also implies that, in the process of using the asymptotic formula, the accuracy of the asymptotic relative error for a higher natural frequency is higher than that for a lower natural frequency. That is to say, in terms of using the asymptotic formula, it is a natural phenomenon that higher natural frequencies converge quicker than lower ones. Therefore, if the accuracy of the natural frequency of mode i is of interest, the asymptotic relative error of the natural frequency of mode i + n can be used to evaluate the possible range of exact error for the natural frequency of mode i. The reason for this is that, in the finite element analysis, the exact error of a lower natural frequency is generally lower than that of a higher natural frequency. In other words, if the asymptotic relative error of the natural frequency of mode i + m (m = 1, 2, . . . , n) is within the given range of error tolerance, the real error of the natural frequency of mode i can be assured within this range. Because of the importance of n, which may be termed as a shift factor, it needs to be very carefully chosen. Generally, the value of n should be large enough so that the real error of the natural frequency of mode i can be guaranteed within a given range of error tolerance. For membrane vibration problems, numerical experience has demonstrated that the value
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of n should be equal to or greater than six for a reasonable finite element mesh and therefore, the value of n is chosen as six in this study. In order to establish a practical error estimator for the finite element solution of a membrane vibration problem, the following two basic questions must be answered. (1) For a given finite element solution, how can its accuracy be evaluated? (2) If the accuracy of a given finite element solution is not satisfactory, how can a satisfactory finite element solution, the real error of which is controlled within the desired range of error tolerance, be obtained for the purpose of a structural design in engineering practice? Since the natural frequency of a dynamic system is a very important characteristic of the system, the accuracy of the finite element predicted natural frequency of a membrane can be used to represent the overall accuracy of the finite element solution for the membrane vibration problem. Therefore, the first criterion of the present practical error estimator in relation to answering question (1) is as follows. For the ith natural frequency predicted from the finite element analysis of a membrane, an upper bound of its real relative error can be provided by the asymptotic relative error of the (i + n)th natural frequency, where n is equal to or greater than six for a reasonable mesh. To answer question two raised above, the adaptive mesh h-refinement scheme is adopted in the finite element analysis since it does not change the element type used but a satisfactory finite element solution is achieved. It has been recognized that, in finite element analysis, the general convergence characteristic of a natural frequency of a system is that, for a linear displacement element, the natural frequency converges as the square of the characteristic length of the elements used. When the characteristic length of the elements used is halved, the error in a specific natural frquency is quartered. This fact leads to the equation: N = (1/ln 4)(ln dir+ n − ln dri ), (15) where dri is the maximum relative error tolerance for the maximum natural frequency, vi of interest, dir+ n is the asymptotic relative error of the natural frequency vi + n and N is the number of times needed to repeat the h-refinement scheme, under the assumption that each existing element is divided into four elements each time. Once the value of N is obtained from equation (15), a new finite element mesh, which results in a finite element solution with accuracy in the given range of error tolerance, can be designed and generated. For example, if N is equal to three, this means that each existing element needs to be divided into 4N elements, namely 64 new elements. It should be pointed out that if N is not an integer then, from equation (15), it is necessary to let N be equal
Figure 1. Finite element model of an L-shaped cantilever membrane (C = 1 m/s).
. . .
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T 1 Asymptotic relative error for the uncorrected natural frequencies (L-shaped membrane) Frequency number
Solution (rad/s)
Mesh 1 3 elements
Mesh 2 12 elements
Mesh 3 48 elements
Mesh 4 192 elements
Mesh 5 768 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
0·186331 0·183118 1·17544 0·442241 0·406983 8·6632 0·588909 0·518206 13·6438 0·721674 0·612805 17·7655 0·866249 0·721038 20·1392
0·178849 0·178112 0·4136 0·396941 0·389225 1·9825 0·538662 0·520196 3·5498 0·718775 0·677878 6·0331 0·827418 0·765609 7·6813 0·923442 0·844769 9·3129 1·118590 0·993074 12·6392 1·169710 1·030520 13·5071 1·226170 1·071350 14·4510 1·311210 1·132070 15·8244
0·176684 0·176505 0·1015 0·385353 0·383514 0·4795 0·519036 0·514590 0·8641 0·671375 0·661899 1·4316 0·775082 0·760685 1·8927 0·830683 0·813091 2·1636 1·003920 0·973665 3·1073 1·033120 1·000310 3·2805 1·110440 1·070230 3·7572 1·191440 1·142500 4·2836
0·175988 0·175943 0·0253 0·382191 0·381738 0·1187 0·514045 0·512946 0·2143 0·659348 0·657037 0·3517 0·761615 0·758065 0·4683 0·805741 0·801544 0·5236 0·971254 0·963949 0·7578 0·990682 0·982936 0·7880 1·073580 1·063760 0·9233 1·129270 1·117870 1·0198
0·175749 0·175735 0·0079 0·381273 0·381159 0·0299 0·512786 0·512513 0·0533 0·656333 0·655760 0·0874 0·758144 0·757260 0·1167 0·799456 0·798420 0·1298 0·962765 0·961167 0·1881 0·979903 0·977998 0·1947 1·064430 1·061990 0·2296 1·113710 1·110920 0·2513
2 3 4 5 6 7 8 9 10
to the smallest integer whose value is greater than the calculated N, for the purpose of assuring the accuracy of the finite element soution from the new mesh. Thus, the second criterion of the present practical error estimator in relation to answering question two can be stated as follows. For a membrane vibration problem, a finite element solution of desired accuracy can be obtained from a new finite element mesh, which is a subdivision of elements used in the original finite element analysis and can be designed by using equation (15).
3. VERIFICATION AND APPLICATION OF THE PRACTICAL ERROR ESTIMATOR
In this section, the practical error estimator is applied to the finite element solution of five representative membrane vibration problems. To reflect the applicability of the present practical error estimator to various problems of different geometric shapes, the domains of these problems are selected as follows: (1) an L-shaped cantilever membrane; (2) a trapezoidal cantilever membrane; (3) a tapered cantilever membrane with a circular hole in the centre; (4) a skewed cantilever membrane; (5) a circular hollow cantilever membrane. For the purpose of verifying the correctness of the practical error estimator and providing a series of finite element solutions for these representative membrane vibration problems,
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the adaptive mesh h-refinement scheme, which results in a monotonic convergence consequence of the finite element solution, is adopted herein. Conventional four-node finite elements are used for the first four problems and eight-node isoparametric elements are used for the fifth problem in the finite element analysis. The wave speed of the membrane is equal to 1 m/s throughout this study. 3.1. - As shown in Figure 1, an L-shaped cantilever membrane is initially divided into three square finite elements of equal areas. In this case, the characteristic length of the elements is equal to the side length of the elements used. Based on the adaptive mesh h-refinement scheme, the domain of the L-shaped cantilever membrane is further subdivided into 12, 48, 192 and 768 square elements. Since all finite elements used in the process of discretization are of equal areas, the characteristic lengths of the elements are 5, 2·5, 1·25, 0·625 and 0·3125 m, respectively, when the L-shaped membrane is divided into 3, 12, 48, 92 and 768 elements. Table 1 shows the asymptotic relative error for the originally predicted natural frequencies of the L-shaped cantilever membrane from the finite element analysis. In this table, the ‘‘uncorrected solution’’ is the finite element solution directly obtained from the finite element analysis, whereas the ‘‘corrected solution’’ is the asymptotic solution which is obtained by substituting the finite element solution into the asymptotic formula expressed in the last section. It can be seen from these results that, for any given natural frequency, the asymptotic relative error decreases with the refinement of a finite element mesh. For example, the asymptotic relative error of the fundamental natural frequency is 1·7544% when three elements are used to model the L-shaped membrane, but it is reduced to 0·4136%, 0·1015%, 0·0253% and 0·0079% when 12, 48, 192 and 768 elements are used. Also, for any given mesh, the asymptotic relative error monotonically increases with the increase of the natural frequency number. For mesh 1 (3 elements) as an example, the asymptotic relative error of the predicted natural frequency increases from 1·7544% to 20·1392% when the natural frequency number increases from 1 to 5. This demonstrates that the asymptotic relative error of the highest natural frequency of interest is a controlling factor in the error estimation of a finite element solution for a membrane vibration problem. In order to verify the correctness of the present practical error estimator, the finite element solution obtained from mesh 5 of 768 elements is viewed as the ‘‘exact solution’’ since the true exact solution is not available for the vibration problem of an L-shaped
Figure 2. Finite element model of a trapezoidal cantilever membrane (C = 1 m/s).
. . .
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T 2 Asymptotic relative error for the uncorrected natural frequencies (trapezoidal membrane) Frequency Solution number (rad/s) 1 2 3 4 5 6 7 8 9 10
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
Mesh 5 1024 elements
0·179418 0·177237 1·2312 0·528056 0·483723 9·1651 0·597811 0·537522 11·2160 0·838263 0·711116 17·8800 1·030620 0·858999 19·9792
0·176242 0·175712 0·3018 0·479703 0·469540 2·1645 0·546177 0·531246 2·7757 0·703367 0·673312 4·4638 0·906972 0·847159 7·0604 0·969133 0·897995 7·9219 1·023090 0·941303 8·6887 1·223360 1·095930 11·6275 1·294760 1·149010 12·6853 1·356990 1·1194540 13·5995
0·175376 0·175245 0·0749 0·461881 0·459509 0·5163 0·531306 0·527711 0·6812 0·671175 0·664006 1·0796 0·820996 0·808060 1·6008 0·889689 0·873348 1·8710 1·938925 0·919826 2·0763 1·118700 1·087120 2·9044 1·162640 1·127410 3·1247 1·220640 1·180210 3·4255
0·175154 0·175121 0·0190 0·457230 0·456648 0·1274 0·527627 0·526734 0·1695 0·662458 0·660696 0·2667 0·797502 0·794438 0·3856 0·867375 0·863441 0·4556 0·917829 0·913174 0·5097 1·083700 1·076070 0·7081 1·103730 1·095680 0·7343 1·163520 1·154120 1·8148
0·175098 0·175090 0·0048 0·456056 0·455912 0·0315 0·526709 0·526486 0·0424 0·660221 0·659784 0·0663 0·791675 0·790919 0·0956 0·861655 0·860682 0·1130 0·912661 0·911505 0·1269 1·072520 1·070650 0·1750 1·090520 1·088550 0·1809 1·149490 1·147190 0·2009
cantilever membrane. By using this ‘‘exact solution’’, the real relative error of the finite element predicted natural frequency of the membrane can be calculated from the equation dirr = (vi − viexact )/viexact ,
(16)
where dirr is the ‘‘real’’ relative error of the finite element predicted natural frequency of mode i, vi is the uncorrected natural frequency of mode i and viexact is the ‘‘exact solution’’ for the natural frequency of mode i. If only the accuracy of the fundamental natural frequency is of interest, its ‘‘real’’ relative error, d1rr , can be worked out by using equation (16), and is 6·0211% (namely, d1rr = (v1 − v1exact )/v1exact = (0·186331 − 0·175749)/0·175749 = 6·0211%), 1·7639% (namely, (0·178849 − 0·175749)/0·175749 = 1·7639%), 0·5320% (namely, (0·176684 − 0·175749)/ 0·175749 = 0·5320%) and 0·1360%(0·175988 − 0·175749)/0·175749 = 0·1360%) for the finite element meshes of 3, 12, 48 and 192 elements respectively, whereas the asymptotic relative errors d7r , are 20·1392%, 12·6392%, 3·1073% and 0·7578% for the corresponding meshes. It is noted that the value of d5r is used instead of d7r for mesh 1 because d7r is not available in this case. Since the asymptotic relative error, d7r , is greater than the ‘‘real’’ relative error, d1rr , it is concluded that the ‘‘real’’ relative error of the predicted natural
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frequency, vi , can be estimated by calculating the asymptotic relative error of the uncorrected natural frequency, vi + n , where n is equal to six for membrane vibration problems. This verifies the correctness of the first criterion of the present practical error estimator. Next, the correctness of the second criterion can be investigated by considering the finite element solution obtained from mesh 2. If the fourth natural frequency of the L-shaped membrane is of interest and its maximum ‘‘real’’ relative error tolerance, dr4 , is equal to 1%, the number of times needed to repeat the h-refinement scheme, under the assumption of each element being divided into four elements each time, can be calculated from equation (15) as r N = (1/ln 4)(ln d10 − ln dr4 ) = (1/ln 4)(ln 0·158244 − ln 0·01) = 1·9920.
(17)
Since N is not an integer, N is set to be 2, which is the smallest integer not only less than N. This requires that in order to obtain a finite element solution in which the ‘‘real’’ relative error of the fourth natural frequency is guaranteed with 1%, each existing element in mesh 2 be divided into 16 new elements. This subdivision leads to a target mesh of 192 elements. That is to say, based only on the information obtained from mesh 2 of 12 elements, a new target mesh can be designed to result in a finite element solution of desired accuracy. It is clear that the ‘‘real’’ relative error of the fourth natural frequency obtained from the mesh of 192 elements is 0·4594%, which is indeed smaller than the given maximum ‘‘real’’ relative error tolerance, dr4 = 1%. Therefore, the correctness of the second criterion of the present practical error estimator has been verified by this example. 3.2. In order to examine the correctness of the present practical error estimator when distorted elements are used in the finite element analysis, a trapezoidal cantilever membrane is selected and initially divided into four quadrilateral elements. Figure 2 shows the initial discretization of the trapezoidal cantilever membrane. Since distorted elements are used to model this problem, the equivalent characteristic length of elements needs to be calculated from the formula h = zA/NE ,
(18)
where h is the equivalent characteristic length of the elements used, A is the total area of
Figure 3. Finite element model of a trapered cantilever membrane with a circular hole in the centre (R = 1·5) C = 1 m/s).
. . .
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T 3 Asymptotic relative error for the uncorrected natural frequencies (trapered membrane with a hole) Frequency number
Solution (rad/s)
Mesh 1 8 elements
Mesh 2 32 elements
Mesh 3 128 elements
Mesh 4 512 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%) v11 v¯ 11 r d11 (%)
0·177568 0·176554 1·5739 0·474199 0·456569 3·8614 0·510789 0·489105 4·4332 0·764263 0·700806 9·0549 0·909498 0·812128 11·9894 1·174960 1·003340 17·1046 1·364710 1·140170 19·6944 1·449630 1·206650 20·1366
0·173988 0·173748 0·1381 0·456207 0·451975 0·9363 0·476962 0·473137 1·0219 0·701946 0·687042 2·1694 0·824138 0·800529 2·9491 1·004690 0·963472 4·2780 1·028270 0·984314 4·4657 1·171790 1·108920 5·6690 1·300520 1·217410 6·8272 1·341860 1·251590 7·2125 1·418480 1·314120 7·9412
0·172678 0·172619 0·0343 0·449213 0·448185 0·2294 0·464391 0·463255 0·2451 0·677495 0·673995 0·5192 0·785463 0·780035 0·6958 0·922868 0·914127 0·9562 0·939806 0·930583 0·9911 1·088690 1·074480 1·3223 1·171910 1·154290 1·5268 1·226840 1·206700 1·6691 1·249570 1·228320 1·7297
0·172230 0·172215 0·0087 0·447113 0·446858 0·0572 0·460839 0·460561 0·0603 0·669899 0·669045 0·1277 0·773575 0·772260 0·1702 0·898832 0·896774 0·2295 0·914096 0·911932 0·2373 1·053260 1·049960 0·3147 1·141140 1·136940 0·3691 1·170160 1·165640 0·3880 1·215330 1·210270 0·4183
2 3 4 5 6 7 8 9 10 11
the problem domain and NE is the total number of elements used in the finite element discretization of the problem domain. When the trapezoidal cantilever membrane shown in Figure 2 is divided into 4, 16, 64, 256 and 1024 quadrilateral elements, the equivalent characteristic lengths of the elements are 4·3301, 2·1651, 1·0825, 0·5413 and 0·2706 m, respectively. As was mentioned before, any refinement of a mesh reduces the equivalent characteristic length of the elements and therefore improves the accuracy of the finite element solution. Table 2 shows the asymptotic relative error for the originally predicted natural frequencies of the trapezoidal cantilever membrane when five different finite element meshes are used in the finite element analysis. With the refinement of the mesh, the finite element solution for any natural frequency of the trapezoidal membrane converges to the exact solution from above because the consistent mass matrix of the element was used in the analysis. The general trend of the asymptotic relative error variation with the increase of the natural frequency number and the refinement of the element mesh is the same as
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observed for the vibration of an L-shaped cantilever membrane. Besides, it can be concluded from these results that the asymptotic relative error of the highest natural frequency of interest is a controlling factor in the error estimation of a finite element solution for a membrane vibration problem, even if distorted elements are used in the finite element analysis. From the finite element solution for the trapezoidal cantilever membrane vibration problem, the validity of the present practical error estimator can be demonstrated as follows. With the second natural frequency of the trapezoidal membrane taken as an example, the ‘‘real’’ relative error of this natural frequency can be worked out by using equation (16) and is 15·7875%, 5·1815%, 1·2773% and 0·2574% for the finite element mesh of 4, 16, 64 and 256 elements, respectively, whereas the asymptotic relative errors d8r are 19·9792%, 11·6275%, 2·9044% and 0·7081% for the respective meshes. Note that the value of d5r is used instead of d8r for mesh 1 because d8r is not available in this case. Since the asymptotic relative error, d8r , is greater than the ‘‘real’’ relative error, d2rr , the validity of the first criterion of the present practical error estimator has been demonstrated, even though quadrilateral elements have been used. To examine the validity of the second criterion of the present practical estimator, it is assumed that the third natural frequency
T 4 Asymptotic relative error for the uncorrected natural frequencies (skewed membrane) Frequency Solution number (rad/s) 1 2 3 4 5 6 7 8 9 10
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
Mesh 5 1024 elements
0·157911 0·155930 1·2704 0·351378 0·332135 5·7967 0·517654 0·465458 11·2139 0·628048 0·546404 14·9421 0·689156 0·584071 16·6333 0·785241 0·658443 19·2572
0·154583 0·154106 0·3095 0·332644 0·328030 1·4066 0·517428 0·500943 3·2908 0·544078 0·525085 3·6171 0·716437 0·675897 5·9980 0·768633 0·719716 6·7967 0·892127 0·820045 8·7900 1·001740 0·905407 10·6397 1·012170 0·913353 10·8191 1·092810 0·973996 12·1986
0·153589 0·153471 0·0769 0·327157 0·326028 0·3463 0·501238 0·497227 0·8067 0·517937 0·512543 0·8573 0·675490 0·665844 1·4487 0·719136 0·707556 1·6366 0·829235 0·811731 2·1564 0·907110 0·884452 2·5618 0·923205 0·899377 2·6494 1·018780 0·987238 3·1950
0·153300 0·153270 0·0196 0·325759 0·325478 0·0863 0·496563 0·495572 0·2000 0·510233 0·509157 0·2113 0·663556 0·661200 0·3563 0·705261 0·702437 0·4020 0·807122 0·802903 0·5255 0·877943 0·872528 0·6206 0·895983 0·890231 0·6461 0·980426 0·972915 0·7720
0·153213 0·153205 0·0052 0·325408 0·325337 0·0218 0·495321 0·495074 0·0499 0·508562 0·508296 0·0523 0·660516 0·659931 0·0886 0·701721 0·701019 0·1001 0·801477 0·800432 0·1306 0·870511 0·869174 0·1538 0·888922 0·887498 0·1538 0·970677 0·968826 0·1911
750
. . .
Figure 4. Finite element model of a skewed cantilever membrane (C = 1 m/s).
of the trapezoidal membrane is of interest and that its maximum ‘‘real’’ relative error tolerance, dr3 , is equal to 1%. By using equation (15) and the information from mesh 2, the value of N is worked out as 1·8325. Therefore, the value of N is taken as 2 and each element in mesh 2 should be divided into 16 new elements, which results in a new target mesh of 256 elements. The ‘‘real’’ relative error of the third natural frequency obtained from mesh 4 of 256 elements is 0·1743%, which is smaller than the given maximum ‘‘real’’ relative error tolerance, dr3 = 1%. This demonstrates that, in spite of using distorted elements in the finite element analysis, the second criterion of the present practical error estimator is also valid for the error estimation of the finite element solution of membrane vibration problems. 3.3. This is a challenging problem for checking the correctness and applicability of the present practical error estimator, since highly distorted elements must be used to model the medium around the circular hole of the membrane. As shown in Figure 3, a tapered cantilever membrane with a circular hole in the centre is initially divided into eight quadrilateral elements. In the adaptive mesh h-refinement scheme, the domain of the
Figure 5. Finite element model of a circular hollow membrane (C = 1 m/s).
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tapered cantilever membrane is further subdivided into 32, 128 and 512 quadrilateral elements. The equivalent characteristic lengths of the elements are 2·9703, 1·4813, 0·7402 and 0·3700 m for the finite element meshes of 8, 32, 128 and 512 elements, respectively. Table 3 shows the asymptotic relative error for the originally predicted natural frequencies of the tapered cantilever membrane with a central circular hole when four different finite element meshes are used in the finite element anlaysis. The finite element solution from mesh 4 of 512 elements is viewed as the ‘‘exact solution’’ since the asymptotic relative error for the fundamental natural frequency is only 0·0087% and the maximum asymptotic relative error for the first 11 natural frequencies is less than 0·5%. The same conclusions as described in sections 3.1 and 3.2 can be drawn from these numerical results shown in Table 3. With the second natural frequency of the tapered membrane taken as an example, the ‘‘real’’ relative error of this natural frequency, d2rr , can be worked out by using equation (16) and is 6·0580%, 2·0340% and 0·4697% for the finite element meshes of 8, 32 and 128 elements, respectively, whereas the respective asymptotic relative errors d8r are 20·1366%, 5·6690% and 1·3223%. This further confirms the validity of the first criterion of the present practical error estimator. If the maximum natural frequency of interest is the fifth natural frequency of the tapered membrane and its maximum ‘‘real’’ relative error tolerance, dr5 , is equal to 2%, the value of N is worked out as 0·9947 by means T 5 Asymptotic relative error for the uncorrected natural frequencies (circular hollow membrane) Frequency number
Solution (rad/s)
Mesh 1 4 elements
Mesh 2 16 elements
Mesh 3 64 elements
Mesh 4 256 elements
1
v1 v¯ 1 d1r (%) v2 v¯ 2 d2r (%) v3 v¯ 3 d3r (%) v4 v¯ 4 d4r (%) v5 v¯ 5 d5r (%) v6 v¯ 6 d6r (%) v7 v¯ 7 d7r (%) v8 v¯ 8 d8r (%) v9 v¯ 9 d9r (%) v10 v¯ 10 r d10 (%)
0·172966 0·171236 1·0103 0·351115 0·337769 3·9512 0·503138 0·467743 7·5672 0·741935 0·650229 14·1036 0·775812 0·674558 15·0104 1·079370 0·898625 20·1135 1·141560 0·958408 19·1100
0·169294 0·168880 0·2451 0·345006 0·341573 1·0051 0·491546 0·481885 2·0048 0·676516 0·652473 3·6849 0·737379 0·706800 4·3264 0·923588 0·867132 6·5107 0·969824 0·905563 7·0962 1·109940 1·018790 8·9469 1·143740 1·045420 9·4048 1·224140 1·107690 10·5129
0·167592 0·167491 0·0603 0·342173 0·341317 0·2508 0·486853 0·484408 0·5047 0·667303 0·661082 0·9410 0·730811 0·722681 1·1250 0·908295 0·892936 1·7201 0·947490 0·930127 1·8667 1·104970 1·077910 2·5104 1·127470 1·098800 2·6092 1·160670 1·129520 2·7578
0·166803 0·166779 0·0144 0·340922 0·340711 0·0619 0·484749 0·484139 0·1260 0·664877 0·663309 0·2364 0·729409 0·727341 0·2805 0·906514 0·902562 0·4379 0·944246 0·939784 0·4748 1·104570 1·097460 0·6479 1·125130 1·117620 0·6720 1·153760 1·145670 0·7061
2 3 4 5 6 7 8 9 10
752
. . .
Figure 6. Vibration modes of the L-shaped cantilever membrane.
of equation (15) and the information from mesh 2. Therefore, the value of N is taken as 1 and each element in mesh 2 should be divided into 4 new elements, which leads to a new target mesh of 128 elements. Clearly, the ‘‘real’’ relative error of the fifth natural frequency obtained from the mesh of 128 elements is 1·5368%, which is smaller than the given maximum ‘‘real’’ relative error tolerance, dr5 = 2%. This demonstrates that, although highly distorted elements have been used in the finite element analysis, the present practical error estimator is a valid error estimation of a finite element solution for the natural frequencies of membrane vibration problems. 3.4. This problem is selected to make a further investigation into the correctness and applicability of the present practical error estimator. As shown in Figure 4, a skewed cantilever membrane is initiallly discretized into 4 quadrilateral elements. In the adaptive mesh h-refinement scheme, the domain of the skewed cantilever membrane is further subdivided into 16, 64, 256 and 1024 quadrilateral elements. The equivalent characteristic lengths of the elements used are 5, 2·5, 1·25, 0·625 and 0·3125 m for the finite element meshes of 4, 16, 64, 256 and 1024 elements, respectively. Table 4 shows the asymptotic relative errors for the originally predicted natural frequencies of the skewed cantilever membrane with five different meshes. For this problem, the finite element solution from mesh 5 of 1024 elements is taken as the ‘‘exact solution’’ since the asymptotic relative error for the first fundamental natural frequency is only 0·0052% and the maximum asymptotic error for the first 10 natural frequencies
Figure 7. Vibration modes of the trapezoidal cantilever membrane.
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Figure 8. Vibration modes of the trapered cantilever membrane with L circular hole in the centre (R = 1·5).
is less than 0·2%. By following the same procedures as used in the first three examples, the validity of both the first and the second criterion of the present practical error estimator is confirmed by the numerical results shown in Table 4. This demonstrates that the present error estimator is applicable to membranes of various geometries. 3.5. This is a curvilinear membrane vibration problem. For this problem, eight-node isoparametric elements are used to model the curved geometry of the membrane. As shown in Figure 5, a circular hollow cantilever membrane is initially divided into four eight-node isoparametric elements. By using the adaptive mesh h-refinement scheme, the circular hollow membrane is further subdivided into 16, 64 and 256 eight-node isoparametric elements. The equivalent characteristic lengths of the elements used are 4·0612, 2·0306, 1·0153 and 0·5077 m for the finite element meshes of 4, 16, 64 and 256 elements respectively. Table 5 shows the asymptotic relative error for the originally predicted natural frequencies of the circular hollow cantilever membrane when four different meshes are used in the finite element analysis. In this case, the finite element solution from mesh 4 of 256 eight-node isoparametric elements is viewed as the ‘‘exact solution’’ because the asymptotic relative error for the fundamental natural frequency is only 0·0144% and the maximum asymptotic relative error for the first 10 natural frequencies is less than 0·8%. The validity
Figure 9. Vibration modes of the skewed cantilever membrane.
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Figure 10. Vibration modes of the circular hollow membrane.
of the present practical error estimator can be verified from the numerical results shown in Table 5, obtained by the same procedures as before. This demonstrates that the present error estimator can be applied to curvilinear membrane vibration problems. Since the aforementioned five representative problems have been studied in great detail, they may serve as benchmark problems, against which any new numerical solution could be compared. For this reason and for the sake of completeness, the first six vibration modes of the L-shaped cantilever membrane, the trapezoidal cantilever membrane, the tapered membrane with a central circular hole, the skewed cantilever membrane and the circular hollow cantilever membrane are shown in Figures 6–10 respectively. All the vibration modes are exaggerated in these figures because, otherwise they are too small to be seen. It can be seen from these figures that the vibration mode of a membrane depends not only on the natural frequency number, but also on the particular shape of the membrane. Generally speaking, for a specific membrane, the larger the natural frequency number, the more complicated the shape of the vibration mode.
4. CONCLUSIONS
The concept of asymptotic error and a practical error estimator for predicted natural frequencies of a membrane vibration problem in the finite element analysis have been presented. The practical error estimator contains the following two criteria: (1) For the ith natural frequency predicted from the finite element analysis, an upper bound of its real relative error can be provided by the asymptotic relative error of the (i + n)th natural frequency, where n is equal to or greater than six for a reasonable mesh of a membrane; (2) for a membrane vibration problem, a finite element solution of desired accuracy can be obtained from a new finite element mesh, which is a subdivision of elements used in the original finite element analysis and can be designed by using equation (15). By using the first criterion, the accuracy of a finite element solution for the natural frequencies of a membrane vibration problem can be evaluated, and by using the second criterion, a target finite element mesh which produces a finite element solution of desired accuracy, can be designed. The correctness and applicability of the present practical error estimator have been demonstrated by the numerical results from vibration problems of five representative membranes of different shapes.
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ACKNOWLEDGMENTS
The authors are grateful to the referees for their valuable suggestions and comments on an earlier draft of this paper.
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