김경종
32 KiB
tends to accumulate round-off errors. A simple example of the decay of diagonal coefficients appears in Fig. 2.11-1. Another example appears in connection with Eq. 18.2-3, where Gauss elimination of u_{1} produces the reduced coefficient K_{22} = (k_{1} + k_{2}) - k_{1} . There it is seen that the reduced K_{22} may have no accurate digits if k_{1} >> k_{2} and computer words are too short. The reduced K_{22} , now perhaps greatly in error, is used as a pivot in eliminating u_{2} , causing error to propagate throughout the solution.
A simple test of accuracy is to check the amount of decay of each K_{ii} before it acts as a pivot. Let P_{ii} be the reduced value of K_{ii} just before it is used as a pivot in eliminating the ith d.o.f. The diagonal decay ratio
r _ {i} = \frac {K _ {i i}}{P _ {i i}} \tag {18.4-1}
where K_{ii} is the original diagonal coefficient, is an approximate measure of the number of digits of accuracy lost; for example, roughly six accurate digits have been lost if any r_{i} is 10^{6} , leaving only p - 6 accurate digits if computer words contain p digits.
Table 18.4-1 shows numerical results obtained from a beam problem on a computer that carries slightly more than eleven digits in double precision arithmetic. We see that it is best if node numbers progress from the free end to the fixed end. (Unfortunately, this arrangement is not practical for all finite element structures.) We also see that trouble may not be detected until reduction is almost complete: in the case where node N is free, trouble appears only in the next-to-last equation, when the equation solver detects that node N is unsupported.
The example in Table 18.4-1 uses elements of equal length. Consider the case N = 1000, now with elements mixed in length but varying in length only between 0.99999 and 1.00001. Then the tip-to-root numbering gives w = 1.0269 while diagonal decay ratios are essentially unchanged from their values of 8.0 and 2.0. This example shows the possibility of substantial accuracy loss that is not detected by the diagonal decay ratio.
If a large coefficient appears only on the diagonal of [K], without corresponding
TABLE 18.4-1. EFFECT OF NUMBER OF BEAM ELEMENTS AND NODE NUMBERING ON ACCURACY, WHERE DIAGONAL DECAY RATIO r_{i} IS DEFINED BY EQ. 18.4-1 AND w IS THE RATIO OF THE COMPUTED DEFLECTION OF LOAD P TO ITS DEFLECTION ACCORDING TO BEAM THEORY.
| N Elements $^{a}$ 2N Nonzero D.O.F. | N Elements $^{a}$ 2N Nonzero D.O.F. | |||||||
| EI = 8(10) $^{6}$ Length = 1000 |  |  |  |  | ||||
| Number of elements | w | $r_{2N-2}$ | $r_{2N-1}$ | $r_{2N}$ | w | $r_{2N-2}$ | $r_{2N-1}$ | $r_{2N}$ |
| N = 10 | 1.0000 | 8.0 | 2.0 | 8.0 | 1.0000 | 5.3 | 1(10) $^{3}$ | 4(10) $^{1}$ |
| N = 100 | 1.0000 | 8.0 | 2.0 | 8.0 | 0.9993 | 7.7 | 1(10) $^{6}$ | 4(10) $^{2}$ |
| N = 1000 | 1.0000 | 8.0 | 2.0 | 8.0 | 0.1197 | 8.0 | 2(10) $^{8}$ | 2(10) $^{3}$ |
^{a} Elements are of equal length (see text).
large off-diagonal coefficients, it does not lead to large diagonal decay. Thus, k_{2} >> k_{1} is acceptable in Eq. 18.2-3, and \beta = 0 or \beta = \pi / 2 is acceptable in Fig. 18.2-3a.
It is the decay of diagonals that is significant, not their smallness. Small pivots per se do not provoke large error. The causes of a large decay ratio r_{i} are the causes of ill-conditioning: inadequate supports, mechanisms, isolated stiff regions, and so on. It is interesting that the condition number of [K] is the same for both beams in Table 18.4-1, roughly 10^{12} when N = 1000, yet the severe loss of accuracy predicted by this large C(\mathbf{K}) materializes for only one of the two beams.
18.5 RESIDUALS
Consider the equation
\{\Delta \mathbf {D} \} = [ \mathbf {K} ] ^ {- 1} \{\Delta \mathbf {R} \} \quad \text { where } \quad \{\Delta \mathbf {R} \} = \{\mathbf {R} \} - [ \mathbf {K} ] \{\mathbf {D} \} \tag {18.5-1}
where \{\Delta R\} is called the residual. If the equations [K]\{D\} = \{R\} could be solved exactly, that is, so that the solution algorithm does not contaminate \{D\} with any round-off error, then \{\Delta R\} = \{0\} . Actually, there is some contamination, and \{\Delta D\} can be regarded as a measure of the probable round-off error in \{D\} . A similar scalar error measure is
e = \frac {\{\mathbf {D} \} ^ {T} \{\Delta \mathbf {R} \}}{\{\mathbf {D} \} ^ {T} \{\mathbf {R} \}} \tag {18.5-2}
Physically, e is the ratio of work done by residual loads to work done by actual loads when both act through displacements \{D\} . If |e| is (say) 10^{-8} or less, round-off contamination is probably negligible [18.7]. However, if the equations [K]\{D\} = \{R\} are ill conditioned, it is possible that inaccurate solutions will yield small residuals and a small e.
A measure of error in vibration analysis, similar to Eq. 18.5-2, is
e = \frac {\{\overline {{{\mathbf {D}}}} \} ^ {T} ([ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \} - \omega^ {2} [ \mathbf {M} ] \{\overline {{{\mathbf {D}}}} \})}{\{\overline {{{\mathbf {D}}}} \} ^ {T} [ \mathbf {K} ] \{\overline {{{\mathbf {D}}}} \}} \tag {18.5-3}
in which the quantity in parentheses would be zero if frequency \omega and mode shape \{\overline{\mathbf{D}}\} were uncontaminated by round-off error.
One can use Eq. 18.5-1 as an iterative improvement scheme. Thus the computed \{\Delta \mathbf{D}\} is added to the existing \{\mathbf{D}\} , a new \{\Delta \mathbf{R}\} is formed, a new \{\Delta \mathbf{D}\} is computed and added to the current \{\mathbf{D}\} , and so on. If this method is to succeed, |e| must be less than unity, and the \{\mathbf{R}\} and [\mathbf{K}] used to compute \{\Delta \mathbf{R}\} must be represented with greater precision than the [\mathbf{K}]^{-1} used to compute \{\Delta \mathbf{D}\} . With [\mathbf{K}] represented accurately and [\mathbf{K}]^{-1} represented inaccurately, convergence is toward the exact solution of [\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\} , as is desired, rather than toward the exact solution of \{\mathbf{D}\} = [\mathbf{K}]^{-1}\{\mathbf{R}\} . Iterative improvement reduces the rounding error introduced during equation solving but does not reduce the inherited error present at the outset of equation solving.
If a set of equations is seriously ill conditioned, it is usually better to rework the finite element model so as to improve its condition than to make heroic attempts
to improve a poor solution by iteration. “If a thing is not worth doing, it is not worth doing well” [18.8].
Interpretation of \{\Delta R\} . A small residual does not guarantee that equations have been solved accurately. Of two approximate solutions, it is possible that the solution of lesser accuracy will yield the smaller residuals [18.8].
In structural mechanics, a small \{\Delta R\} indicates that applied loads \{R\} are balanced by resisting loads [K]\{D\} arising from the deformed structure. Checking that \{\Delta R\} \approx \{0\} is sometimes called a “statics check” or an “equilibrium check.” Passing an equilibrium check is no guarantee that results are accurate. For example, one may compute deflections and stresses that are physically unrealistic because the mesh is too coarse, yet find \{\Delta R\} \approx \{0\} (merely because the equation solver is working properly).
In summary, small residuals are a necessary but not sufficient condition for accuracy. Reference 18.21 suggests a residual norm test: that \|\Delta R\|<<\|\widetilde{R}\| if the solution is sufficiently accurate. Reference 18.21 also suggests that the solution is probably acceptable if either the condition number C(\mathbf{K}) alone or the residual norm test alone indicates trouble, but probably unacceptable if both C(\mathbf{K}) and the residual norm test indicate trouble.
18.6 DISCRETIZATION ERROR:
ANALYSIS
In representing a mathematical continuum by finite elements, we select the number, type, and shape of elements, the grading of the mesh, allocate distributed loads to nodes, and represent support conditions by fixing certain d.o.f.. The approximation inherent in this process is called discretization error. Unfavorable discretization can provoke subsequent numerical difficulty, for example, as described in Section 18.2. However, in the present section we are concerned only with discretization error itself, that is, with the discrepancy between the discretized model and the mathematical model, with the latter being taken as correct.
A simple rule is that the volume of the discretized structure should be correct. Thus, if a circular region is modeled by a polygon of straight-sided elements, the polygon should neither inscribe nor circumscribe the circle; rather, element sides should intersect the circle so that the area of the polygon equals the area of the circle. Similarly, straight elements that model an arch should not be chords, but longer, so that the sum of element lengths equals the arch length.
Error Analysis. Discretization error can sometimes be determined by an order of error analysis. Consider, for example, the axially loaded bar of Fig. 18.6-1a. From the differential element, the equation of axial equilibrium is A\sigma_{x,x} + q = 0 . Or, substituting the stress–strain relation \sigma_{x} = Eu_{,x} , the equilibrium equation is
u _ {, x x} + \frac {q}{A E} = 0 \tag {18.6-1}
Let the discretized model consist of standard two-d.o.f. bar elements. It is not hard to show that if the load integral is evaluated consistently (i.e., by Eq. 4.1-6), and if A and E are constant, then finite element displacements are exact
at the nodes (see Problems 18.28 and 18.29). This does not mean that finite element displacements are exact everywhere: the exact solution is in general not piecewise linear, so there is error between the nodes.
The behavior of the error in the bar model can be understood by doing a comparatively simple error analysis. The error of the bar model is in many ways either characteristic of, or very relevant to, the behavior of the error in more complicated and realistic problems. We cannot hope to obtain discretization error bounds in all practical problems. But many model problems can be analyzed, each reflecting salient features of a more practical problem. What we touch on here is most directly applicable to linear elasticity problems with smooth (but possibly varying) elastic coefficients, with distributed (not point) loading, in bodies whose geometry does not induce strong singularities (no cracks). The arguments and conclusions rest on a fundamental fact of finite element error analysis: for a sufficiently refined mesh, the error in the finite element solution can be bounded by the error in approximating the exact solution by shape function interpolation that is exact at nodes [18.9]. Thus, whether or not the finite element solution is exact at nodes (and in general it is not), the error in nodal interpolation tells the story. The proviso of “a sufficiently refined mesh” is intended to exclude models having large discretization error, such as a beam modeled by a few constant-strain triangles, whose computed nodal displacements are greatly in error.
Accordingly, in the bar problem we analyze the error of linear interpolation between exact nodal displacements u_{i} and u_{i+1} . In the ith element, displacement error e = e(x) is
e (x) = u (x) - u _ {i} \left(1 - \frac {x - x _ {i}}{h _ {i}}\right) - u _ {i + 1} \left(\frac {x - x _ {i}}{h _ {i}}\right) \tag {18.6-2}
where u(x) is the exact solution. Also, u_{i} = u(x_{i}) and u_{i+1} = u(x_{i+1}) , where x_{i} and x_{i+1} are the nodal locations. Element length is h_{i} = x_{i+1} - x_{i} . There is no need to require all element lengths to be equal. In the i th element, error e(x) is a smooth function. At nodes, there are discontinuities in the first derivative e'(x) , as shown by Fig. 18.6-1c. We note that e(x_{i}) = e(x_{i+1}) = 0 .
One can determine an upper bound on e(x) by the following argument [18.9]. Let z be an axial coordinate x_{i} \leq z \leq x_{i+1} such that e'(z) = 0 . That there is such a point is obvious from Fig. 18.6-1b and follows rigorously from Rolle's theorem of elementary calculus. In what follows we assume that u''(x) is continuous in the interval x_{i} \leq x \leq x_{i+1} . The change in e' can be computed by integrating e'' . Thus, with s a dummy variable, e'(z) = 0 , and using Eq. 18.6-2,
e ^ {\prime} (x) - e ^ {\prime} (z) = e ^ {\prime} (x) = \int_ {z} ^ {x} e ^ {\prime \prime} (s) d s = \int_ {z} ^ {x} u ^ {\prime \prime} (s) d s \tag {18.6-3}
because the linear terms in Eq. 18.6-2 do not contribute to e''(x) . In addition,
\left| \int_ {z} ^ {x} u ^ {\prime \prime} (s) d s \right| \leq \int_ {z} ^ {x} | u ^ {\prime \prime} (s) | d s \leq \int_ {x _ {i}} ^ {x _ {i + 1}} | u ^ {\prime \prime} (s) | d s \leq h _ {i} \left(\max _ {x _ {i} \leq x \leq x _ {i + 1}} | u ^ {\prime \prime} (x) |\right) \tag {18.6-4}
^{1} In the remainder of this section we use the notation e_{,x} = e' , e_{,xx} = e'' , and so on.
text_image
Exact solution, u = u(x) Finite element (piecewise linear) x_{i+1} x_i x_i dx σ_x A (σ_x + dσ_x)A q dx q = q(x) i i + 1 x ith element
(α)
text_image
e(x) x_i x_{i+1} h_i O(h_i^2) x x z s (b)
text_image
e'(x) = \frac{d}{dx}e(x) O(h_i) (c) x
Figure 18.6-1. The behavior of the error in a uniform bar under distributed axial load q. The exact axial displacement is u = u(x) and the error is e = e(x) .
Therefore, on the ith element the error in strains e'(x) is bounded by
\left| e ^ {\prime} (x) \right| \leq h _ {i} \left(\max _ {x _ {i} \leq x \leq x _ {l + 1}} \left| u ^ {\prime \prime} (x) \right|\right) \tag {18.6-5}
We can also bound the displacement error e(x) by observing that it must have greatest magnitude at x = z, where e'(z) = 0 . Now z must be closer to x_i or to x_{i+1} . Assume that z is closer to x_i , and compute e(x_i) by a three-term Taylor series, with exact remainder, expanded about x = z,
e (x _ {i}) = e (z) + (x _ {i} - z) e ^ {\prime} (z) + \frac {1}{2} (x _ {i} - z) ^ {2} e ^ {\prime \prime} (s) \tag {18.6-6}
where s is the remainder evaluation point on element i . But e(x_{i}) = 0 , e'(z) = 0 , and e''(s) = u''(s) from Eq. 18.6-2, so
e (z) = - \frac {1}{2} \left(x _ {i} - z\right) ^ {2} u ^ {\prime \prime} (s) \tag {18.6-7}
According to the assumption that z is closer to x_{i} than to x_{i + 1} , we conclude that |z - x_{i}| \leq h_{i} / 2 . Therefore, the error in displacements, e(x) , is bounded by
e (x) \leq \frac {1}{8} h _ {i} ^ {2} \left(\max _ {x _ {i} \leq x \leq x _ {i + 1}} \left| u ^ {\prime \prime} (x) \right|\right) \tag {18.6-8}
One may verify that the same result is obtained when z is assumed to be closer to x_{i+1} than to x_i .
We note that the existence of a z for which the error in strain is zero is the rationale for the existence of the optimal stress calculation points discussed in Section 6.13.
Important features of the foregoing discussion are as follows.
- The strain error is proportional to element size, and the displacement error is proportional to the square of element size.
- The error estimates are proportional to derivatives one order higher than the degree of the shape functions (for the bar, second derivatives and linear shape functions are involved).
- Displacements are most accurate at or near element nodes. Strains are most accurate in element interiors, that is, at or near Gauss points.
Using the notation “O” for “order,” we say that Eq. 18.6-5 displays a discretization error O(h) in strain and Eq. 18.6-8 displays a discretization error O(h^{2}) in displacement. Thus, if h = \max(h_{i}) is cut in half to produce two elements, h is halved; the error in strain is approximately halved and the error in displacement is approximately quartered.
Remarks. We would like to generalize the foregoing conclusions to other elements and to strain energy error as well [18.10]. For this purpose we define symbols as follows:
h = approximate "characteristic length" of an element: length of a linear element; length of the longest line segment connecting two points in a plane or solid element
q - 1 = degree of highest complete polynomial in the element displacement field
2m = order of highest derivative in the governing equilibrium equation expressed in terms of displacements
For the bar of Fig. 18.6-1, q - 1 = 1 and 2m = 2 . For the standard four-d.o.f. beam element, q - 1 = 3 and 2m = 4 . In plane and solid problems with bilinear and trilinear elements discussed in Chapter 6, q - 1 = 1 and 2m = 2 . For bending of thin plates with twelve d.o.f. elements based on Eq. 11.2-5, q - 1 = 3 and 2m = 4 .
Because an element can fit exactly a displacement field of degree q - 1, it therefore has error O(h^{q}) in representing the polynomial fields of degree q and higher that in general are present in the exact solution. The error in strains (and
therefore stresses) is proportional to the error in the r th derivative of the displacement field; that is, the stress error is O(h^{q-r}) . For plane and solid elements, r = 1 . For beam and plate elements, for which stresses are dictated by curvatures, r = 2 . Thus the standard four-d.o.f. beam element has displacement error O(h^{4}) and stress error O(h^{2}) .
In plane problems it is sometimes possible to use optimal points for stress calculation, as noted in this section and in Section 6.13. Then the stress error is O(h^{q}) rather than O(h^{q-1}) .
A strain energy expression contains squares of the mth displacement derivatives. Therefore, the strain energy error is O(h^{2q-2m}) .
Again we remark that the foregoing error estimates are based on the assumption that nodal loads are evaluated consistently, that is, by Eq. 4.1-6. If ad hoc load lumping is used, or if Eq. 4.1-6 is evaluated by too low a quadrature rule, the estimates may not apply. For example, the beam deflection data seen at the end of Section 4.3 use ad hoc load lumping and display displacement error O(h^{2}) rather than the expected O(h^{4}) .
Singularities. It is a general principle in finite element error estimation that constants involved in an error bound are proportional to the qth partial derivative of the exact solution. If this derivative is infinite, the error bound does not rigorously apply. For example, in problems with cracks, displacement derivatives through the second are singular at the crack tip; therefore, even with linear elements (for which q - 1 = 1), our estimates do not apply. Other problems may have singularities in higher derivatives. We often do not know the order of the lowest singular derivative, and therefore we cannot estimate the factor by which error will be reduced by subdividing the mesh.
However, although singularities may slow convergence, the presence of singularities in derivatives higher than the second does not rule out convergence. A polynomial of degree q - 1 necessarily contains also a polynomial of degree q - p for 1 < p \leq q . If the (q - p + 1) -order derivative of the exact solution is the highest-order derivative that is nonsingular, then we can apply our previous estimates with q - 1 replaced by q - p to deduce that the error in displacements is O(h^{q-p+1}) , the error in strains (calculated from first derivatives) is O(h^{q-p}) , and so on. The implication of this is that for each derivative of order q or less that is singular in the exact solution, we risk losing a power of h accuracy in the finite element solution. Such an estimate may be pessimistic, and often is. But when it holds, elements of degree q - p will have the same order of error as elements of degree q - 1, and the lower-degree elements will have less computational expense.
As an example, consider again the axially loaded bar, with a step change in loading within an element. Specifically, for the element that spans x = x_{i} to x = x_{i+1} , let x_{i} < x_{q} < x_{i+1} and
q = 0 \text { for } x \leq x _ {q} \quad \text { and } \quad q = 1 \text { for } x > x _ {q} \tag {18.6-9}
Then, from Eq. 18.6-1 u_{,xx} is defined for all x , \max |u''(x)| = 1/AE , and Eqs. 18.6-5 and 18.6-8 make sense. However, u'''(x) is a delta function at x = x_q , so \max |u'''(x)| is undefined. Accordingly, the error estimates remain as stated in Eqs. 18.6-5 and 18.6-8, whether the element is linear, quadratic, or of yet higher degree: the O(h^3) displacement accuracy one would normally expect from a quadratic
element is not available. Full accuracy of a quadratic element could be recovered by placing a node at the step change in load, for example, set x_{i} = x_{q} or set x_{i+1} = x_{q} .
For C^{0} elements in general, we do not expect to lose much accuracy when properties such as A, E, v, thickness, and so on, display sudden jumps at inter-element boundaries. But if properties jump within elements or if the exact solution displays “blow-up” singularities in higher derivatives, considerable accuracy may be lost.
18.7 DISCRETIZATION ERROR:
ESTIMATION AND EXTRAPOLATION
In the latter part of Section 18.6 we argued that error e in a computed quantity could be represented as e = O(h^{q-r}) , where r = 0 for displacement error, r = 1 for stress or strain error in elasticity, and so on. A restatement of this result is
e \approx C h ^ {q - r} \tag {18.7-1}
where h is again the “characteristic length” and C is a problem-dependent constant that is influenced by element aspect ratio (ratio of longest side to shortest side), size ratio of the largest and smallest elements in the mesh, element type, quadrature rule, the (q - p + 1) -order derivative of the exact solution (as in Section 18.6), and other factors [18.19]. Error e in a computed quantity \phi may indeed refer to the error at a point, but more generally is
e = \left\| \phi - \phi^ {h} \right\| \tag {18.7-2}
where \phi is the exact value, \phi^h is the computed value, and the norm symbol (\| \cdot \|) indicates an averaging process; for example, e may represent the root-mean-square of several pointwise values, or an integrated average over a portion of the body, or the square root of strain energy in the entire body, and so on. If a single value of \phi is intended, then e = |\phi - \phi^h| .
The relative error is e_r = \| \phi - \phi^h \| / \| \phi \| . A rough indicator of this error is
e _ {r} \approx \rho_ {1} \rho_ {2} h ^ {q - r} \quad \text { where } \quad h = \frac {1}{N ^ {1 / n}} \tag {18.7-3}
Here q = one plus the degree of the highest complete polynomial in the element displacement field, r is defined above Eq. 18.7-1, \rho_{1} = largest element aspect ratio, \rho_{2} = ratio of characteristic length of the largest element to characteristic length of the smallest element, h = “dimensionless length,” N = number of elements in the mesh, and n = spatial dimension (n = 1, 2, or 3 for line, plane, and solid problems, respectively). Thus h is the characteristic length of an element in a domain that has been scaled so that its length, area, or volume is unity. The analyst has some control over all quantities in Eq. 18.7-3 except n. Our interpretation of e_{r} is as follows. If e_{r} is about 0.1 times the acceptable percentage error, the results are probably reliable (e.g., e_{r} = 1\% if 10% error is acceptable). If e_{r} is considerably smaller than this, the mesh may already be finer than necessary. If e_{r} approaches or exceeds unity, further study is required. “Further study” may
radar
| x | y | Value |
|---|---|---|
| 1 | 28 | 10 |
| 1 | 19 | 11 |
| 1 | 10 | 12 |
| 1 | 37 | 20 |
| 1 | 46 | 36 |
| 1 | 55 | 27 |
| 1 | 64 | 18 |
| 1 | 73 | 36 |
| 1 | 82 | 27 |
| 2 | 28 | 10 |
| 2 | 19 | 11 |
| 2 | 10 | 12 |
| 2 | 37 | 20 |
| 2 | 46 | 36 |
| 2 | 55 | 27 |
| 2 | 64 | 18 |
| 2 | 73 | 36 |
| 2 | 82 | 27 |
| 3 | 28 | 10 |
| 3 | 19 | 11 |
| 3 | 10 | 12 |
| 3 | 37 | 20 |
| 3 | 46 | 36 |
| 3 | 55 | 27 |
| 3 | 64 | 18 |
| 3 | 73 | 36 |
| 3 | 82 | 27 |
| 4 | 28 | 10 |
| 4 | 19 | 11 |
| 4 | 10 | 12 |
| 4 | 37 | 20 |
| 4 | 46 | 36 |
| 4 | 55 | 27 |
| 4 | 64 | 18 |
| 4 | 73 | 36 |
| 4 | 82 | 27 |
| 5 | 28 | 10 |
| 5 | 19 | 11 |
| 5 | 10 | 12 |
| 5 | 37 | 20 |
| 5 | 46 | 36 |
| 5 | 55 | 27 |
| 5 | 64 | 18 |
| 5 | 73 | 36 |
| 5 | 82 | 27 |
| 6 | 28 | 10 |
| 6 | 19 | 11 |
| 6 | 10 | 12 |
| 6 | 37 | 20 |
| 6 | 46 | 36 |
| 6 | 55 | 27 |
| 6 | 64 | 18 |
| 6 | 73 | 36 |
| 6 | 82 | 27 |
| 7 | 28 | 10 |
| 7 | 19 | 11 |
| 7 | 10 | 12 |
| 7 | 37 | 20 |
| 7 | 46 | 36 |
| 7 | 55 | 27 |
| 7 | 64 | 18 |
| 7 | 73 | 36 |
| 7 | 82 | 27 |
| 8 | 28 | 10 |
| 8 | 19 | 11 |
| 8 | 10 | 12 |
| 8 | 37 | 20 |
| 8 | 46 | 36 |
| 8 | 55 | 27 |
| 8 | 64 | 18 |
| 8 | 73 | 36 |
| 8 | 82 | 27 |
| 9 | 28 | 10 |
| 9 | 19 | 11 |
| 9 | 10 | 12 |
| 9 | 37 | 20 |
| 9 | 46 | 36 |
| 9 | 55 | 27 |
| 9 | 64 | 18 |
| 9 | 73 | 36 |
| 9 | 82 | 27 |
| 10 | 28 | 10 |
| 10 | 19 | 11 |
| 10 | 10 | 12 |
| 10 | 37 | 20 |
| 10 | 46 | 36 |
| 10 | 55 | 27 |
| 10 | 64 | 18 |
| 10 | 73 | 36 |
| 10 | 82 | 27 |
Figure 18.7-1. A “Laplacian” plane mesh. Coordinates x_{i} and y_{i} of each interior node are equal to the average of the coordinates of the four adjacent nodes, for example, x_{11} = (x_{2} + x_{10} + x_{12} + x_{20})/4 .
mean mesh refinement, alteration of the mesh without changing the number of elements, or assessment of the physical reasonability of the results if computational limitations do not permit mesh refinement or rearrangement.
However, Eq. 18.7-3 may be very pessimistic. One can easily devise a patch test in which results are exact yet e_{r} > 1000 . Clearly, the role of stress gradients is not addressed by Eq. 18.7-3. In preparing or refining a mesh, an analyst must draw upon experience and intuition about gradients present and how well they can (or should) be modeled: if a test quantity such as stress or strain energy changes appreciably across interelement boundaries or within individual elements, mesh refinement may be needed.
Consider, for example, Fig. 18.7-1. Here N = 72, n = 2, and we estimate \rho_{1} = 2 and \rho_{2} = 8 . Therefore, if elements are bilinear (q = 2), we estimate e_{0} \approx 0.22 for displacements (r = 0) and e_{1} \approx 1.9 for stresses (r = 1). These estimates indicate that results may not be reliable. This may indeed be the case if we seek the effect of a local disturbance, say a concentrated force at node 36. On the other hand, if nodes along the left edge are fixed and nodes along the right edge are loaded by a uniform traction, displacements and stresses near node 81 would probably be reliable because the mesh is smoothly graded, finer where stress gradients are likely to be larger, and fine enough at the location of interest. A uniform mesh ( \rho_{1} = \rho_{2} = 1 ) with elements the size of the smallest element in Fig. 18.7-1 might improve accuracy slightly, but would display e_{0} = O(10^{-3}) and e_{1} = O(10^{-2}) , and would be much more costly. We conclude that by mesh grading and suiting the mesh to the problem, one can “beat the estimate.”
Multimesh Extrapolation. It is possible to use Eq. 18.7-1 and the computed results from two or more different meshes to extrapolate to an improved result. Ideally, this result has zero error. What rigor there is in this process depends on completely regular mesh refinement. That is, in each refinement, nodes and interelement boundaries of the coarser mesh are preserved, both in number and location, while adding new nodes and new boundaries; \rho_{1} and \rho_{2} of Eq. 18.7-3 remain constant; element types and quadrature rules are not changed; corner nodes stay corner nodes and side nodes stay side nodes (Fig. 18.7-2). In addition, the quantity to
natural_image
Pure grid diagram with dashed lines and dots, no text or symbols present
{a}
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Pure grid diagram with dashed lines and dots, no text or symbols present
(b)
Figure 18.7-2. Dashed lines indicate new element boundaries introduced by regular mesh subdivision. New nodes introduced by subdivision are not shown. (a) Bilinear elements; divisions by 2 shown. (b) Quadratic elements; divisions by 3 shown.
be extrapolated must be calculated at a location fixed in space and fixed in position relative to an element (e.g., at a corner node) in all refinements. If such regularity is lacking, extrapolation can still yield a useful estimate, but without a sound basis in theory.
If only two meshes are used, and the convergence rate is unknown, we have little choice but to assume that convergence is linear ( e = O(h) ; see Fig. 18.7-3a). With data from three meshes, we may conclude that convergence is indeed linear, or that quadratic (or higher) convergence is a possibility. A too-coarse mesh may fail to display a definite trend. Incompatible elements often display nonmonotonic convergence (curve AD in Fig. 18.7-3b). With nonmonotonic convergence, two meshes yield no reliable prediction: clearly, two data points can be extrapolated to a result that is better, worse, or no different, depending on the points chosen. If convergence is monotonic and p is the order of error, e = O(h^p) , then an improved result \phi^0 is obtainable by Richardson extrapolation [18.11],
\phi^ {0} = \frac {\phi_ {1} h _ {2} ^ {p} - \phi_ {2} h _ {1} ^ {p}}{h _ {2} ^ {p} - h _ {1} ^ {p}} \tag {18.7-4}
where, as examples, p = 1 for e = O(h) in Fig. 18.7-3a and p = 2 for e = O(h^2) in Fig. 18.7-3b. Equation 18.7-4 usually gives an inexact \phi^0 because conditions
line
| h | φ (Curve A) | φ (Curve B) | | ---- | ----------- | ----------- | | 0 | φ₀ | φ₀ | | h₂ | e = O(h) | e = O(h) | | h₁ | φ₁ | φ₁ |{a}
line
| h² | φ (Curve A) | φ (Curve B) | φ (Curve C) |
|---|---|---|---|
| 0 | φ⁰ | φ² | φ¹ |
| h² | φ² | φ¹ | φ² |
| h² | e = O(h²) | φ¹ | φ² |
{b}
Figure 18.7-3. Convergence of a quantity \phi with mesh refinement. AB, linear convergence. AC, quadratic convergence, AD, nonmonotonic convergence (but quadratic convergence when h < h_{D} ).