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An adaptive numerical integration algorithm for polygons / C. J. Li; C. Dagnino. - In: APPLIED NUMERICAL MATHEMATICS. - ISSN 0168-9274. - STAMPA. - 60:2009(2010), pp. 165-175.
Original Citation:
An adaptive numerical integration algorithm for polygons
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DOI:10.1016/j.apnum.2009.11.001
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Chong-Jun Li, Catterina Dagnino
An adaptive numerical integration algorithm for polygons
Appl. Numer. Math., 60 no.3 (2010), 165-175, DOI 10.1016/j.apnum.2009.11.001
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An adaptive numerical integration algorithm for
polygons
✩Chong-Jun Li∗,a, Catterina Dagninob a
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
b
Department of Mathematics, University of Torino, via C. Alberto, Torino 10123, Italy
Abstract
In this paper an adaptive numerical integration algorithm for polygons is constructed by cubatures based on finite quadrilateral elements, with partic-ular reference to an 8-node spline finite element. Some numerical examples are given to evaluate the performance of the proposed algorithm, combined with several cubatures with accuracy of order two. The results are also com-pared with those obtained by two cubatures based on triangulation and a recent method based on the Green’s integral formula and Gaussian rules. Key words: Adaptive algorithm; Numerical cubature; Spline finite element; Convergence; Triangulated quadrangulation
1. Introduction
The problem considered in this paper is the adaptive evaluation of inte-grals in the form
IΩ(f ) = Z
Ω
f (x, y)dxdy, (1)
where Ω is a polygon domain in R2 with boundary composed of piecewise straight lines.
✩Project supported by the WWS- World Wide Style Project of University of Torino
(Fondazione CRT) (Italy), Science Foundation of Dalian University of Technology (China) (No. SFDUT07001).
∗Corresponding author.
Email addresses: chongjun@dlut.edu.cn(Chong-Jun Li), catterina.dagnino@unito.it(Catterina Dagnino)
We can evaluate (1) by subdividing the domain Ω into triangular or quadrilateral elements, then applying a cubature formula on each element and finally summing up the values obtained.
Some adaptive integrators, based on the subdivision of Ω into triangles, were presented in [3, 5]. Different to subdivision, Green’s integral formula is used explicitly in the numerical cubature context in [12]. A kind of Gauss-like cubature formulas over polygons is constructed by transforming a two-dimensional into a one-two-dimensional problem and by using univariate Gauss quadratures, that we will denote by GR. The cubature can obtain a very good approximation of integrals of smooth functions, since it converges as the degree of accuracy increases. However, it is not efficient for integrand functions with singularity inside the integral domain as shown in [12]. More-over, if the domain Ω is not convex or with holes, then some GR cubature nodes fall outside Ω.
Besides, some papers also focus on numerical algorithms based on quad-rangulations ([4, 11]). The authors mentioned that in some situations for both the finite element and the scattered data interpolation problems, it is preferable that the finite elements be quadrangles (quadrilaterals) instead of triangles. For example, it has recently been shown that quadrangulations have several advantages over triangulations for the problem of scattered data interpolation [8] and that improvements in elasticity analysis can be obtained in finite element methods by using quadrangles rather than trian-gles [1]. Hence, here we are interested to a subdivision of Ω into quadri-lateral elements Q1, Q2, . . . , QN. If ˜IQk(f ) is a cubature rule defined on Qk, k = 1, . . . , N , then we can write
IΩ(f ) = N X k=1 Z Qk f (x, y)dxdy ≃ ˜IΩ(f ) = N X k=1 ˜ IQk(f ). (2) There are two questions for getting (2). The first one is the choice of a suitable cubature ˜IQk(f ), the second one is the strategy of the domain subdivision.
With reference to the first question, a tensor product cubature with a few nodes could be applied after a bilinear transformation of any quadrilateral domain into a rectangular one. That is, let Ai (xi, yi), i = 1, 2, 3, 4 be four vertices of the quadrangle Q, the four Lagrange bases be
N1 = 1 4(1 − u)(1 − v), N2 = 1 4(1 + u)(1 − v), N3 = 1 4(1 + u)(1 + v), N4 = 1 4(1 − u)(1 + v).
By the bilinear transformation x = 4 X i=1 xiNi, y = 4 X i=1 yiNi, (3)
one integral on quadrangle Q is transformed to another integral on [−1, 1]2, i.e., Z Q f (x, y)dxdy = Z 1 −1 Z 1 −1
f (x(u, v), y(u, v))|J(u, v)|dudv, (4) where the Jacobian |J(u, v)| is a bilinear polynomial of u and v.
However, the degree of accuracy will be reduced because of the trans-formation. For example, if f (x, y) is a polynomial of total degree 2 with respect to x and y, then f (x(u, v), y(u, v)) is a biquadratic polynomial of u and v. So, the integrand in (4) on [−1, 1]2 is a bicubic polynomial including the Jacobian. It means that a cubature is only exact for quadratic polyno-mials on quadrilateral domain though it is exact for bicubic polynopolyno-mials on rectangular domain.
The 2×2 Gauss-Legendre cubature with four nodes or the tensor product Simpson formula could seem suitable choices. Both have a degree of accuracy 3 on a rectangular element and only 2 on a quadrilateral domain. We denote by G4 and S9, respectively, the corresponding global formulas for (2). For G4 the total number of cubature nodes is 4N and it rapidly increases when the number of elements increases, since all nodes are interior of each element. The S9 cubature has one node inside each quadrilateral element.
Since it could be interesting to have nodes located only on the boundary of the quadrilateral elements, we can use the L8-cubature rule introduced and studied in [10]. It is based on the 8-node quadrilateral spline element presented in [9]. Such rule has the same degree of accuracy of G4 and S9, but its total node number is less than the G4’s and S9’s one on the same quadrangulation, because each node of L8 is shared by several elements.
With reference to the second question, here we propose a procedure similar to that one given in [7], where a globally adaptive algorithm uses successive refinements or subdivisions of Ω, where each subdivision is used to provide a better approximation to I(f ). These subdivisions are designed to dynamically concentrate the computational work in the subregions of Ω where the integrand f is more irregular and thus to adapt to the behavior of the integrand. The general structure of the algorithm in [7] consists of a sequence of stages, with each stage composed by the following five main steps:
1) Select a subregion with the largest estimated error from the current set of subregions;
2) divide the selected subregion;
3) apply a local cubature rule to any new subregion; 4) update the subregion set;
5) update the global integral and error estimates, and check for termination. The first two steps are keys for the above adaptive algorithm. In order to es-timate the error on each subregion, the difference of values obtained by two different local cubature rules is adopted in [7]. Moreover the direction to di-vide a region is along the integrand irregularity by computing the difference of integrand values.
In this paper, only one local cubature rule is used in the proposed adap-tive algorithm. The error estimate is computed by the difference of the integral approximation in two successive steps. In particular, firstly any subregion is subdivided, then the subregion with the largest error estimate is selected to be subdivided again.
The main steps of our algorithm are the following: 1) Divide the initial subregions;
2) apply a local cubature rule to any new subregion; 3) update the subregion set;
4) select a subregion with the largest error estimate from the current set of subregions;
5) update the local cubature values and error estimates, and check for ter-mination.
The paper is organized as follows. In Section 2 the L8 cubature is re-viewed. Section 3 presents the adaptive algorithm. Finally in Section 4 some numerical tests are proposed, with comparisons with the results obtained by using GR, G4, S9 rules and two cubatures based on triangulation.
2. The cubature based on the 8-node quadrilateral spline finite element
Suppose that ♦ is a nondegenerate convex quadrangulation of a closed polygonal domain Ω in R2, and ♦ is composed of N convex quadrangles Qk, k = 1, 2, . . . , N . Let ∆Q be the triangulation of ♦ generated by adjoin-ing both diagonals of each quadrangle, as shown in Fig. 1.
Fig. 1: A triangulated quadrangulation.
For each convex quadrangle Q, denote the four vertices and four mid-points on each edge by A1, · · · , A8, and denote the intersection of two diag-onals A1A3 and A2A4 by A0, as shown in Fig. 2. The quadrangle is divided into four subtriangles ∆1, . . . , ∆4. The areas of the four subtriangles are
S1= 1 2 1 x0 y0 1 x1 y1 1 x2 y2 , S2 = 1 2 1 x0 y0 1 x2 y2 1 x3 y3 , S3 = 1 2 1 x0 y0 1 x3 y3 1 x4 y4 , S4 = 1 2 1 x0 y0 1 x4 y4 1 x1 y1 ,
where a, b, c, d are defined by a = |A4A0|
|A4A2|
, b = |A3A0| |A3A1|
, c = 1 − a, d = 1 − b. (5)
In [9] an 8-node spline element (denoted by L8) has been defined by the spline basis of a quadratic spline space on ∆Q ([13, 14]). Then, a numerical cubature on polygons, based on the above L8-element, has been proposed in [10]. It has a degree of accuracy two for an arbitrary convex quadrangle and three for a rectangle or parallelogram domain.
1 2 3 4 5 6 7 8 4 A 3 A 2 A 1 A 1 ∆ 2 ∆ 3 ∆ 4 ∆ A0
Figure 2: A convex triangulated quadrangle and its 8 boundary nodes.
In this case the cubature formula on a quadrilateral element Q has the following form ([10]) ˜ IQ(f ) = 8 X i=1 Cif (Ai). (6) where C1= − 1 6b(S1+ S2+ S3+ S4); C2= − 1 6a(S1+ S2+ S3+ S4); C3= − 1 6d(S1+ S2+ S3+ S4); C4= − 1 6c(S1+ S2+ S3+ S4); C5= 1
3((1 + a + b + ab)S1+ (b + ab)S2+ abS3+ (a + ab)S4); C6=
1
3((d + ad)S1+ (1 + a + d + ad)S2+ (a + ad)S3+ adS4); C7= 1 3(cdS1+ (c + cd)S2+ (1 + c + d + cd)S3+ (d + cd)S4); C8= 1 3((c + bc)S1+ bcS2+ (b + bc)S3+ (1 + b + c + bc)S4), (7) with a, b, c, d given by (5). The sum of absolute values of the cubature coefficients is bounded as follows
8 X i=1 |Ci| = 5 3meas(Q), (8)
where meas(Q) denotes the area of Q. Hence the cubature (6) is stable ([10]).
In ([10]) the convergence of sequences of cubatures (2), with ˜IQ(f ) de-fined by (6) and (7), is proved and error bounds are obtained. In particular, we have
|IΩ(f ) − ˜IΩ(f )| = o(δj), if f ∈ Cj(Ω), 0 ≤ j ≤ 2,
where δ denotes the length of the longest diagonal or edge in the quadran-gulation ♦ of Ω.
3. The adaptive algorithm AC1
1 G G2 3 G 4 G (a) G4 1 A A2 3 A 4 A 5 A 6 A 7 A 8 A (b) L8 1 A A2 3 A 4 A 5 A 6 A 7 A 8 A A9 (c) S9
Figure 3: The location of nodes for G4, L8, S9 cubatures on a quadrilateral element.
The input for our adaptive algorithm is given by: the integrand function, the vertices of the initial quadrilateral elements of Ω, and a tolerance ε. Step 1 INITIALIZATION. Denote Q1, Q2, . . . , QN the initial quadrilateral
elements.
1) For each element Qk, k = 1, 2, . . . , N ,
a. compute the cubature value ˜Ik = ˜IQk(f ) by (6);
b. divide Qk into two new elements Qk,1 and Qk,2 by defining a new edge according to the following strategy:
b1) For the G4 cubature, if
max(|f1− f2|, |f3− f4|) ≥ max(|f1− f4|, |f2− f3|), then the new edge is A5A7, else the new edge is A6A8, where fi = f (Gi), i = 1, . . . , 4, (as shown in Fig. 3). b2) For the L8 cubature, if
max(|f5− f1|, |f5− f2|, |f7− f3|, |f7− f4|) ≥ max(|f6− f2, |f6− f3|, |f8− f4|, |f8− f1|), then the new edge is A5A7, else the new edge is A6A8, where fi = f (Ai), i = 1, . . . , 8, (as shown in Fig. 3).
b3) For S9 cubature, if
max(|f9− f6|, |f9− f8|) ≥ max(|f9− f5, |f9− f7|), then the new edge is A5A7, else the new edge is A6A8, where fi = f (Ai), i = 1, . . . , 9, (as shown in Fig. 3). c. compute ˜Ik,j= ˜IQk,j(f ), j = 1, 2 by (6);
d. compute the absolute value of the cubature differences, i.e. ∆ ˜Ik= | ˜Ik− ( ˜Ik,1+ ˜Ik,2)|;
2) Let ∆ = 1, and S = {Q1, Q2, . . . , QN} the set of selected ele-ments. Denote the cardinal number of S by ♯S.
Step 2 LOOP. While ∆ ≥ ε/♯S, do
1) for each element Qj ∈ S, j = 1, 2, . . . , ♯S,
a. replace Qj by the sub-elements Qj,1 and Qj,2, i.e. let N = N + 1, Qj = Qj,1 and QN = Qj,2. Update the cubature values by the two new elements, i.e., ˜Ij = ˜Ij,1 and ˜IN = ˜Ij,2; b. divide the new element Qj into two sub-elements Qj,1 and
Qj,2, then compute ˜Ij,1, ˜Ij,2 and ∆ ˜Ij = | ˜Ij− ( ˜Ij,1+ ˜Ij,2)|; c. divide the new element QN into two sub-elements QN,1 and
QN,2, and compute ˜IN,1, ˜IN,2and ∆ ˜IN = | ˜IN−( ˜IN,1+ ˜IN,2)|. 2) let S = ∅, then compare all ∆ ˜Ik, k = 1, 2, . . . , N , and compute ∆ = maxk=1,...,N{∆ ˜Ik}. Add the elements with maximum differ-ence into the set S.
Step 3 When
∆ < ε/♯S, (9)
1) for each element Qj (j = 1, . . . , N ), let N = N + 1, Qj = Qj,1, QN = Qj,2, ˜Ij = ˜Ij,1 and ˜IN = ˜Ij,2.
2) compute the sum of cubature values on each element, i.e. ˜IΩ(f ) = PN
k=1I˜k. The output of the algorithm is the global cubature value ˜
IΩ(f ).
The global cubature value is only computed in the last step. We re-mark the local error estimate ∆ = maxk=1,...,N{∆ ˜Ik} is used to check for termination. The global error estimate can be obtained easily by the local one. Indeed denote by h the number of the loop in Step 2 and by N (h) the number of quadrilateral elements at the end of the h-th loop. Then the
global cubature value of the h-th loop is ˜IΩ(h)(f ) =PN(h)
k=1 I˜k. Note that only the elements in S = {Qh1, . . . , Qh♯S} are divided in the next loop, hence
| ˜IΩ(h+1)(f ) − ˜IΩ(h)(f )| ≤ ♯S X
j=1
∆ ˜Ihj = ♯S · ∆.
When the termination condition (9) is satisfied, the global error estimate | ˜IΩ(h+1)(f ) − ˜IΩ(h)(f )| < ε.
From Fig. 3, it is clear that total number of cubature nodes is V + E for L8 and V + E + N for S9, compared with 4N for G4, where V is the number of the vertices of the quadrangulation, E is the number of the edges, respectively.
The algorithm above described has been implemented in Matlab and the code can be requested to the first author.
4. Numerical examples
In this section, some numerical examples are presented to test the algo-rithm AC1.
The test domain is shown in Fig. 4 with five initial quadrilateral elements. The coordinates of the eleven vertices are (0, 0.75), (0.25, 0.5), (0.25, 0), (0.75, 0.5), (0.75, 0), (1, 0.5), (0.75, 0.75), (1.0, 0.9), (0.5, 1), (0.875, 0.625), (0.5, 0.75). The test functions we considered are the following:
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11
Figure 4: A non-convex domain with initial quadrilateral elements.
f2(x, y) = (x + y)19, f3(x, y) = e−100((x−0.5) 2+(y−0.5)2) , f4(x, y) =p|x − y|, f5(x, y) = e− (5−10x)2 2 + 0.75e− (5−10y)2 2 + 0.75e− (5−10x)2 2 − (5−10y)2 2 , f6(x, y) =p(x − 3/5)2+ (y − 3/5)2, f7(x, y) = |(x − 1/2)2+ (y − 1/2)2− 1/16|, f8(x, y) =p|3 − 4x − 3y|.
The reference integral values have been computed by Mathematica NIn-tegratefunction with 20-digit WorkingPrecision [15], as shown in Table 1.
Table 1: Reference integral values of the test functions, computed by Mathematica
f IΩ(f ) f1 0.15767705312825664 f2 1100.7845363608947 f3 0.031220839802646214 f4 0.24629044427941857 f5 0.3558027665056683 f6 0.15538983599176648 f7 0.0354330513939807 f8 0.5169123732639505
Table 2 shows the numerical results by the adaptive algorithm AC1 based on L8 cubature, where ’EleN’ denotes the number of elements, ’NodN’ de-notes the number of nodes, ’Rel-Err’ is the relative error compared by the reference value, and ’Tol’ is the tolerance ε.
For all test functions, the Fig. 5 shows the meshes and nodes involved when AC1, based on L8 cubature, stops. We can remark the cubature elements and nodes are clustered according to the place where the integrand is irregular.
All test functions have been also integrated both by the GR cubature and by G4, L8, S9 cubatures, using both an equal subdivision technique and the adaptive algorithm AC1. The Fig. 6-9 show the graphs of the corresponding relative errors. The x-axis denotes the number of function values (or cubature nodes), labelled by ’PTS’. In the graphs the line labelled ’GR’ denotes the relative error by GR formula, the lines labelled ’G4’, ’L8’, ’S9’ denote the errors by G4, L8, S9 rules with equal subdivision strategy,
Table 2: Cubature relative errors for the test functions, computed by L8 rule
f EleN NodN Rel-Err Tol
f1 616 2177 1.10E-006 1.0E-8 f2 777 2606 2.89E-006 1.0E-8 f3 1669 5610 2.70E-006 1.0E-8 f4 1570 6171 1.94E-006 1.0E-8 f5 1287 4510 1.57E-006 1.0E-8 f6 584 2069 1.37E-006 1.0E-8 f7 1323 5394 8.21E-007 1.0E-8 f8 1689 6554 2.82E-006 1.0E-8
and the lines ’Ad-G4’, ’Ad-L8’, ’Ad-S9’ those by G4, L8, S9 rules with the adaptive algorithm AC1. In the last case, we set the tolerance ε = 10−6 for AC1. We can remark that when the termination condition (9) is satisfied, the adaptive algorithms based on G4, L8 and S9 stop with a different number of nodes, as shown in the figures.
The figures show that the results by adaptive algorithm are better than those by equal subdivision technique, since AC1 needs less nodes to obtain the same order of errors, especially for not smooth functions f1, f4, f6, f7, f8. For smooth functions f2, f3, f5, the errors by GR cubature are better than those by AC1 and equal subdivision methods. However, GR cubature nodes fall outside the non-convex test domain. Therefore, the integrand function f has to be computed also in the rectangular domain containing the polygon and the error estimate involves the best uniform polynomial approximation on such rectangular domain, as mentioned in the Remark 2.4 of [12].
In addition, the numerical results are also compared by two cubatures based on triangulation, denoted by T3 and T37. T3 is the standard rule of quadratic polynomials using 3 midpoints at the 3 edges of a triangle. T37 is another basic rule of polynomials of degree 13 using 37 points adopted in [3]. The rule can be found in [2] or [6]. The two cubatures are applied with the adaptive algorithm AC1 (modified with triangulation). In this case, the triangle with the largest error estimate is selected to be subdivided into four subtriangles. The results are shown in Fig. 6-9 by lines ’Ad-T3’ and ’Ad-T37’, respectively. For smooth functions f2, f3, f5, ’Ad-T37’ works well because of its high degree of accuracy. However, for not smooth functions f1, f4, f6, f7, f8, it is not better than other low degree cubatures.
Finally we remark the algorithm AC1 has the following advantages: 1) it is suitable for integrand functions with low order smoothness, because
it can distinguish the place where the integrand is not regular.
2) the cost of computation and the number of cubature points are saved greatly by avoiding equal subdivision.
Acknowledgements
The authors thank the University of Torino for its support to WWS-World Wide Style Project.
The first author also thanks the National Natural Science Foundation of China (Nos. 60533060, 10726067) and the Natural Science Foundation for Doctoral Career of Liaoning Province of China (No. 20061060) for their support to his research.
We also thank the reviewers for some constructive suggestions and com-ments.
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0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) f1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) f2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (c) f3 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (d) f4 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (e) f5 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (f) f6 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (g) f7 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (h) f8
0 200 400 600 800 1000 1200 1400 1600 1800 10−7 10−6 10−5 10−4 10−3 10−2 10−1 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (a) f1 0 200 400 600 800 1000 1200 1400 1600 1800 10−15 10−10 10−5 100 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (b) f2
0 200 400 600 800 1000 1200 1400 1600 1800 10−8 10−6 10−4 10−2 100 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (a) f3 0 200 400 600 800 1000 1200 1400 1600 1800 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (b) f4
Figure 7: The comparison for the relative errors of approximation integral values for f3
0 200 400 600 800 1000 1200 1400 1600 1800 10−12 10−10 10−8 10−6 10−4 10−2 100 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (a) f5 0 200 400 600 800 1000 1200 1400 1600 1800 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (b) f6
0 200 400 600 800 1000 1200 1400 1600 1800 10−8 10−6 10−4 10−2 100 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (a) f7 0 200 400 600 800 1000 1200 1400 1600 1800 10−7 10−6 10−5 10−4 10−3 10−2 10−1 PTS Relative Error GR G4 L8 S9 Ad−G4 Ad−L8 Ad−S9 Ad−T3 Ad−T37 (b) f8
