Staggered Solution Procedures

Transcription

1 . 7 Staggered Solution Procedures 7 1

2 Chapter 7: STAGGERED SOLUTION PROCEDURES 7 2 TABLE OF CONTENTS Page 7.1. Partition Strategies Staggered Algebraic Partition Implementation Corrective Iteration Staggered Differential Partition Implementation Notes and. Bibliography References

3 STAGGERED ALGEBRAIC PARTITION 7. STAGGERED SOLUTION PROCEDURES 7.1. Partition Strategies Two partition strategies may be used to construct staggered solution procedures. Algebraic Partitioning. The full system is treated with an implicit integration operator as described previously. The resulting algebraic system is partitioned in accordance with a field separation. Finally, a predictor is applied to the state vector that appears on the right hand side. Differential Partitioning. The matrices appearing in the differential system are partitioned. The right hand side is treated with a predictor. Finally, an integrator operator is applied to the predicted differential system. The distinction between these two strategies can be more easily remembered through three letter acronyms paired with the formulation steps. Algebraic partitioning: integrate-partition-predict or IPP. Differential partitioning: partition-predict-integrate, or PPI. Anticipating results, it is emphasized that the two strategies do not lead to the same results. However, for certain predictor choices thay can be made to coalesce. Historically, differential partition was the original approach used in the derivation of staggered solution procedures for fluid-structure interaction, and remains a natural way to handle physically diverse systems. Algebraic partitioning was initially formulated for structure-structure and control-structure interaction to provide additional flexibility of implementation Staggered Algebraic Partition To keep the notation comparatively simple we assume that the same integration operator has been used on both u and v. We therefore start from the algebraic system resulting from the discretization of Mü + C u + Ku = f (t), and split the coefficient matrix as where H = H 1 + H 2, (7.1) H 1 = M + δc 1 = δ 2 K 1, H 2 = δc 2 + δ 2 K 2. (7.2) The term involving H 2 is then transferred to tyhe right-hand side and a predictor is applied: H 1 u n+1 = g n+1 H 2 u P n+1. (7.3) To complete the specification of the time integration procedure, the configuration of the matrices C 1 through K 2 must be defined. For the case of two partitions, with field state vectors xbold and y, a staggered solution procedure is obtained if we select [ ] [ ] Cxx 0 0 Cxy C = C 1 + C 2 = +, C yx C yy [ Kxx 0 K = K 1 + K 2 = K yx K yy ] [ 0 Kxy 0 0 Note that the mass matrix M is not partitioned. It must be stressed, however, that the staggered partition is only one a myriad of possible partitions. This numberes increases rapidly. For three fields the number of possible matrix partitions rises to 120, and to 2520 for four fields [7.20]. 7 3 ]. (7.4)

4 Chapter 7: STAGGERED SOLUTION PROCEDURES Implementation Having decided upon a staggered solution procedure, it remains to work out the implementation details. By this it is meant the selection of auxiliary vectors, computational paths, and predictor formula. The latter two can be be crucial to the success of the solution method, whereas the former affects computational efficiency and ease of programming of tasks such as starting and stepsize changing procedures. As regards the choice of auxiliary vector and computational paths, the discussion of Chapter 6 applies with only formal changes. These changes pertain to the mechanics of going through calculation sequences such as those illustrated in that Chapter, and breaking down the computations in accordance with the partitions (7.4). To fix the ideas, Figure 7.1 details the calculation sequence associated with the (J0) formulation. Recall that this identification corresponds to pairing the computational path (0) with the (J) choice of auxiliary vector v. Since u is partitioned into x and y, v is partitioned into two subvectors p and q as per the definition [ ] p v = = M u + Cu. (7.5) q In partitioned form this becomes p = M xx ẋ + C xx x + C xy y, q = M yy ẏ + C yx x + C yy y. (7.6) In Figure 7.1, the computations pertaining to the x and y fields are placed on the right and left columns, respectively. Steps associated with single field (decoupled field) analysis calculations are enclosed in solid line boxes. Steps required on account of field coupling are displayed in dashed boxes. Thus Figure 7.1 helps to answer the question; given a specific partition and computational path, what extension to the single-field analysis programs are necessary? In this respect, Figure 7.1 is more detailed than the largely conceptual advancing diagrams presented in Chapter 2. For formulations other than (J0) only a few computational steps in the upper half of Figure 7.1 have to be modified. The reader is referred to Tables 6.1 and 6.2 for details. Note that the question: which predictor? has not been answered. Decisions in this respect must wait until the next Chapter. Therein we shall see that the selection of computational path and predictor are strongly linked Corrective Iteration A corrective iteration cycle may be implemented in Figure 7.1 (and also in Figure 7.2) by re-entering the predict box on the right column, and replacing yn+1 P with the last computed y n+1. This is called a corrective iteration, which can be obviously extended to more cycles. This may be useful, for instance, if the y field contains hard nonlinearities, for example in free-boundary, moving interface or contact problems. For linear and midly nonlinear problems, it has been our experience than cutting the time step is more effective than iterating. There is a notable exception, however. If the response of the coupled mechanical system is dominated by rigid body motions, a staggered partition treatment may induce significant accuracy degradation on x-field components near the partition boundary [7.62]. In such case, a one-cycle iteration may result in significant accuracy improvement at modest cost. Another way 7 4

5 STAGGERED DIFFERENTIAL PARTITION x field r xy n = K xy y n y field n n + 1 ṗ n = fn x K xxx n rn xy p n = h p n 1 + δ pṗ n hn p = h m β p ṗ n i+1 m α p p n i+1 hn x = h m β x ẋ n i+1 m i=1 αx i x n i+1 gn+1 x = M xxhn x + δ phn p + δ pδ x fn+1 x H xx = M xx + δ p C xx + δ p δ x K xx q n = fn y K yyy n rn yx q n = h q n 1 + δ q q n h q n = h m q i=1 βq i n i+1 m hn y = h m ẏ n i+1 m β y g y n+1 = M xxh y n + δ qh q n + δ qδ y f y n+1 H yy = M yy + δ q C yy + δ q δ y K yy q i=1 αq i n i+1 i=1 αy i y n i+1 n n + 1 x n+1 = H 1 xx (gx n+1 gxy n+1 ) ẋ n+1 = (x n+1 h x n )/δ x Predict y P n+1 g xy n+1 = (δ qc xy + δ q δ y K xy )y P r yx n+1 = K yxx n+1 g n+1 = δ p C yx x n+1 + δ x δ p r yx n+1 y n+1 = H 1 yy (gy n+1 gyx n+1 ) ẏ n+1 = (y n+1 h y n )/δ y Figure 7.1. (J0) implementation of algebraic partitioned staggered solution procedure. out is to use three-field partitions, in which the boundary between x and y is treated as a separate field. Variants of the three-field partitions, such as the element-by-element variant, preserve exactness of rigid body motions [7.62,7.65] but demand a more complex implementation than staggered procedures Staggered Differential Partition In this approach we start from the governing differential equations. Introduce the splittings C = C 1 +C 2, K = K 1 + K 2 of the damping and stiffness matrices, and transfer the terms involving C 2 and K 2 to the right hand side: Mü + C 1 u + K 1 u = f C 2 u K 2 u. (7.7) The next step is to apply a predictor at t = t n+1 : Mü n+1 + C 1 u n+1 + K 1 u n+1 = f n+1 C 2 u P n+1 K 2 up n+1. (7.8) The right hand side of (7.8) can be viewed as a pseudo- or effective-force vector ˆf n+1 : Mü n+1 + C 1 u n+1 + K 1 u n+1 = ˆf n+1. (7.9) Finally, (7.10) is treated with a LMS integrator. This step produces the difference system H 1 u n+1 = g n+1, (7.10) where, for the case of a common integration formula and v = AM u + Bu, H 1 = M + δc 1 + δ 2 K 1, g n+1 = M δ(c 1 A 1 B)h u n + A 1 h v n + δ2ˆf n+1. (7.11) 7 5

6 Chapter 7: STAGGERED SOLUTION PROCEDURES 7 6 x field y field ṗ n = fn x K xxx n rn xy p n = h p n 1 + δ pṗ n hn p = h m β p ṗ n i+1 m α p p n i+1 hn x = h m β x ẋ n i+1 m i=1 αx i x n i+1 gn+1 x = M xxhn x + δ phn p + δ pδ x fn+1 x q n = f y n K yyy n r yx n q n = h q n 1 + δ q q n h q n = h m i=1 βq i q n i+1 m h y n = h m β y ẏ n i+1 m i=1 αq i q n i+1 i=1 αy i y n i+1 n n + 1 H xx = M xx + δ p C xx + δ p δ x K xx g y n+1 = M xxh y n + δ qh q n + δ qδ y f y n+1 H yy = M yy + δ q C yy + δ q δ y K yy n n + 1 ˆf x n+1 = f n+1 x + f xy n+1 x n+1 = H 1 xx (gx n+1 + δ pδ u f xy x n+1 = (x n+1 hn x )/δ x n+1 ) Predict y P n+1 and ẏp n+1 f xy n+1 = C xyẏ P n+1 K xyy P n+1 r yx n+1 = K yxx n+1 g n+1 = δ p C yx x n+1 + δ x δ p r yx yx n+1 y n+1 = H 1 yy (gy n+1 gyx n+1 ) ẏ n+1 = (y n+1 h y n )/δ y Figure 7.2. (J0) implementation of differential partitioned staggered solution procedure Implementation As in the case of the staggered algebraic partition, the implementation of the time advancing step is completed through the selection of auxiliary vector, computational path and predictors. Figure 7.2 illustrates the calculation sequence involved in the programming of the (J0) formulation. This results from combining path (0) with the auxiliary vector selection [ ] p v = = M u + C q 1 u. (7.12) which in partitioned form is p = M xx ẋ + C xx x (7.13) q = M yy ẏ + C yx x + C yy y. Note that (7.13) is not generally equivalent to the auxiliary vector definition for the algebraic partition unless the coupled damping terms C xy and C yx vanish. The calculation sequence in Figure 7.2 displays more anisotropy in the treatment of the fields than that of Figure 7.1. This results primarily from the pseudo-force treatment; that is, embodying predicted terms in ˆf x, which is later reused in the calculation of ṗ. Another distinctive feature is the need for two predictors, one for y n+1 and one for yn+1 P, whereas in the algebraic staggered procedure only one predictor is necessary. However for many application problems only one of the terms occur. Notes and Bibliography The appearance of staggered solution procedures in what is now called a differential partition form, is outlined in the Notes and Bibliography of Chapter 2. The distinction between algebraic and differential partitions emerged later when other applications such as structure-structure interaction were considered. Further analysis of both partition types may be found in [7.20,7.61,7.62,7.65,7.66]. 7 6

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