The collimation system of LHC will consist of flat collimator jaws distributed along the IR7 lattice with the aim of limiting the maximum combined amplitudes of secondary halo particles (born along the edges of the primary collimators). The code DJ (Distribution of Jaws) computes this amplitude using a quasi-analytic algorithm (no tracking), by which the maximum initial angles are found, corresponding to trajectories escaping all secondary jaws. We report the latest version of DJ, which contains the following enhancements: (1) the orientation of each pair of jaws is a free variable (instead of using only vertical, horizontal, or 45 degrees skew jaws); (2) the minimizing method used is "simulated annealing", which, for our case of a discontinuous function of up to 32 variables, always finds a global minimum. Different initial jaw distributions lead to different final ones, but they all give essentially the same maximum halo amplitude; this seems to depend only on the number of jaws and the lattice parameters, particularly the tune-split. We discuss lattice characteristics found favorable for collimation

Craddock, M K

Jeanneret, J B

Kaltchev, D I

Servranckx, R V

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsLarge Hadron Collider Project LHC Project Report 134Numerical Optimization of Collimator Jaw Orientations and Locations in theLHCD.I.Kaltchev, M.K.Craddock,z, R.V.Servranckx, J.B.Jeannerety,xAbstractThe collimation system of LHC will consist of at collimator jaws distributed along the IR7lattice with the aim of limiting the maximum combined amplitudes of secondary halo particles(born along the edges of the primary collimators). The code DJ (Distribution of Jaws) computesthis amplitude using a quasi-analytic algorithm (no tracking), by which the maximum initialangles are found, corresponding to trajectories escaping all secondary jaws. We report the latestversion of DJ, which contains the following enhancements: (1) the orientation of each pair ofjaws is a free variable (instead of using only vertical, horizontal, or 45 degrees skew jaws); (2)the minimizing method used is "simulated annealing", which, for our case of a discontinuousfunction of up to 32 variables, always nds a global minimum. Dierent initial jaw distributionslead to dierent nal ones, but they all give essentially the same maximum halo amplitude;this seems to depend only on the number of jaws and the lattice parameters, particularly thetune-split. We discuss lattice characteristics found favorable for collimation.Paper presented at the Particle Accelerator Conference, PAC97, Vancouver, Canada,12 May 1997.TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T2A3yCERN{SL DivisionzAlso at Dept. of Physics & Astronomy, UBC, Vancouver, Canada.xjbj@mail.cern.chAdministrative SecretariatLHC DivisionCERNCH-1211 Geneva 23SwitzerlandGeneva, September 19971 INTRODUCTIONThe betatron beam collimation system for the LHC will be installed in the IR7insertion. It will consist of a set of primary collimators, followed by a number of secondarycollimators arranged to limit the so-called secondary beam halo produced at the edgesof the primaries, thereby protecting the LHC vacuum chamber from scattered particles.Each collimator will be composed of a pair of opposing at jaws.In [1] we presented the computer code DJ, which distributes secondary jaws alongthe IR7 lattice with the aim of minimizing the largest surviving combined amplitude ofthe halo. We now report some enhancements to DJ and the improved results which theyhave made possible.2 DEVELOPMENT OF THE MODEL2.1 Varying the jaw anglesIn the code DJ, the production of halo particles is modeled by a set of point-likesources distributed along the borders of the primary jaws (straight lines in the transverseplane). The secondary jaws, assumed to act as black absorbers, are dened by theirhorizontal tune advance x(within the collimation section IR7) and their rotation angle around the longitudinal axis.For a given jaw distribution, a mapping technique is used to isolate the fraction oftrajectories escaping all secondary jaws, i.e. those passing between the two opposing jawsof all pairs. For these uncaptured halo particles, the code nds the maximum combinedx-y betatron amplitude Amax, where A = (A2x+ A2y)1=2and Ax,Ayare the single-planetransverse invariants. This computation is fast (1 s) as no tracking is needed. DJ furtherminimizes Amaxas a function of the jaw distribution vector (x1; x2:::; xN; 1; 2:::; N).In [1] four types of jaws are used { vertical, horizontal and two skew { with rotationangles  = 0; 90; 45, 135respectively. In the new version of the code DJ, the angle  isan independent variable, along with the jaw position, and may range over 0   180;during minimization the jaw positions and angles are varied together. The improvementin Amax(expressed in terms of the r.m.s. amplitude ) is shown in Table 1, for primaryand secondary collimators set at 6 and 7 respectively. For 12 secondary collimators,allowing  to vary has the same eect as adding 4 more secondaries; for 16 secondariesthe improvement is less dramatic.Table 1:Number of AmaxAmaxsecondary pairs discrete  = 0   180of jaws 0; 90; 45; 13512 9.4 8.716 8.6 8.4With variable angles, the program module calculating Amaxfor a xed jaw dis-tribution remains unchanged. However, the minimization of Amaxwith twice as manyvariables called for some new numerical tools and solutions.2.2 MinimizationDuring minimization, the locations of the four primary jaws are xed at maxima ofthe corresponding beta functions, their angles being: 0; 90; 45and 135.1The optimization process has been developed in two stages: 1) conventional meth-ods, which allow better insight, but show an unwelcome dependence on the initial condi-tions, with some runs ending up in local minima, and 2) simulated annealing.Amaxis not a smooth function of the jaw-distribution vector, because of screeningeects by some secondary jaws on others. A step of nite length made in any directionof the 2N -coordinate space may result in unpredictable changes in the indices m;n ofthe two maximum amplitude jaws [1] and correspondingly in Amax. However, for smallenough increments, the coordinates of the two maximum-amplitude jaws are the only oneswhose variables aect Amax, so there are only four active variables m; m; n; n(localsmoothness).Downhill-overstep methods [2] are based on the local smoothness of Amax. At eachiteration, the LMDIF package routine is used with four variables up to the limits of thesmoothness interval. After that, a step is made outside this interval, to pick a new pairof active jaws. The step is halved after each unsuccessful iteration (no downhill directionfound).Simulated annealing (SA) { a probabilistic optimization method [3], is a recenttechnique devised for solving dicult problems involving discontinuous multi-variablefunctions, but requiring large computing time. The Appendix oers a quick overview ofSA in one dimension.At early stages of minimization, if the percentage of accepted cases rises, thenthe range over which the code searches for an optimum increases, i.e. the SA algorithmkeeps more than one local minimum in sight. As the \temperature" parameter is reduced,downhill moves are less likely to be accepted, more cases are rejected and SA focuses onthe global extremum.In several initial runs, appropriate values were chosen for the most important SAparameters { the initial temperature (T0= 5) and the temperature reduction factor (0.6).A typical SA run assumed xed IR7 lattice functions, a suciently large number of sourcepoints along the edges of the four primary collimators, and 12 to 16 secondary collimators.With these parameters xed, SA runs made for random initial jaw distributionsalways resulted in essentially the same minimum value for Amax, as desired. The nal jawdistributions, however, were by no means identical, although many were very similar (seex3.2 below). The secondary-halo cross-sections diered correspondingly, having dierentmaximum single-plane invariants Axmaxand Aymax(but the same amplitude near thediagonal in (Ax; Ay) space).If DJ is modied to search only for jaw locations compatible with the rest of thehardware, then the computing time increases unacceptably. The alternative approach wastaken of shifting the quadrupoles slightly to free locations at which jaws were needed.3 RESULTSDierent IR7 tunes were explored for several recent versions of the LHC lattice, andfor 16 secondary collimators Amaxwas found to be between 8.4 and 9.1, depending onthe lattice setting.3.1 Optimum lattice settingOptics criteria can be formulated in terms of x(s) and y(s) { the horizontal andvertical tune advances along the straight section, with x(s0) = y(s0) = 0 at the rstprimary collimator (s0= 290 m), or equally, in terms of the functions = (yx)=2. The average tune advance +is roughly proportional to the distance from the2Figure 1: IR7 lattice and tune-split functions for LHC version 4.2, with IR7 quadrupolestuned for high negative tune split, giving Amax= 9.1.rst primary: +/ s   s0. Therefore, for a xed length of the collimation section, thecollimation quality can essentially be expressed in terms of the total +and the tune-splitfunction  (s).The advantage of having the tune split vary along the beamline was rst suggested,and conrmed by tracking, by Risselada [4]. For the case of circular collimators, initialstudies have been carried out [5], aiming to explain the relation between the shape of (s) and the collimation quality, and a search for a rigorous theory is under way.As reported in [1], larger oscillations in  (s) give lower Amax, but we have alsofound dependence on the sign of  . The gures below show the lattice and tune-splitfunctions of IR7 for two cases recently studied: a tune giving large negative  and Amax= 9.1 (Fig. 1), and a tune giving large positive  and Amax= 8.45(Fig. 2).The following tune-split variation gave Amax< 8:5:- almost everywhere positive and close to periodic, with three nearly equal maxima 0:2 each (Fig. 2, bottom);- one high peak in the middle  0:25 (an abandoned lattice version, not shown).On the other hand, a tune giving two large negative peaks in  (Figure 1, bottom), givesa somewhat higher Amax= 9.1.3.2 Optimum collimator phasesFor a lattice optimized purely for collimation, some relation is to be expected be-tween the horizontal and vertical betatron phases of perfectly located collimators. Evenin realistic lattices, constrained by additional factors, the SA runs showed that this re-mains true, favouring certain jaw locations. We also found that +is a more relevantindependent variable than s.3Figure 2: IR7 lattice and tune-split functions for LHC version 4.2, with the IR7quadrupoles tuned for high positive tune split, giving Amax= 8.45.Fifty SA runs were performed for nearly optimum conditions (Amax= 8:45) usingthe lattice shown in Fig. 2. Each run used a randomly generated initial jaw distribution,i.e. random angles and phases for 16 collimators. The values of  were then plotted against+(Fig. 3) for the resultant 50 jaw distributions, which all give nearly the same value ofAmax: 8:4 < Amax< 8:5. The jaw locations tend to cluster near the extreme valuesof the function  (+). On the other hand, for the lattice with Amax= 9.1 the patternis more chaotic (Fig. 4).References[1] D. Kaltchev, M.K. Craddock, R.V. Servranckx, J.B. Jeanneret, Optimization of Colli-mator Jaw Locations for the LHC, Proc. EPAC96 (Barcelona, June 1996), ed. S. Myerset al., (IOP, Bristol, 1996), 1432-4.[2] D. Kaltchev, M.K. Craddock, R.V. Servranckx, Progress Report on Optimizing Colli-mator Jaw Angles and Positions with the Code DJ (Distribution of Jaws), TRIUMFdesign note (in preparation).[3] A. Corana, M. Marchesi, C. Martini, S. Ridella, Minimizing multi-modal functions ofcontinuous variables with the "simulated annealing" algorithm, ACM Transactions onMathematical Software Vol. 13, No. 3 (Sept. 1987), 262-280.[4] T. Risselada, Optical Requirements for an LHC Cleaning insertion with EllipticalCollimators, SL Note 95-67, 1995.[5] T. Risselada, private communication.4Figure 3: Collimator distributions from 50 SA runs for the lattice shown in Fig. 2 ( Amax=8.45) using 16 pairs of secondary jaws with random initial settings. Each point representsthe rotation angle and the average betatronic phase +of one pair.4 APPENDIX: EXAMPLE OF SIMULATED ANNEALINGALGORITHM IN ONE DIMENSIONIn this example, the global minimum of F (x) is searched for, within some inter-val (x1; x2). The user supplies initial values for x (x1< x < x2), the step dx and thetemperature parameter T (T > jx1  x2j).At each iteration, 20 trial values xtriare generated randomly in the interval (x  dx; x + dx). If a trial value xtriis downhill, i.e. F (xtri) < F (x), then it is accepted and,if F (xtri) is lower than its previously lowest value, xtriis recorded as a new optimum.An uphill xtrican also be accepted with probability P = e(F (x) F (xtri))=T(Metropoliscriterion). If the trial xtriis out of bounds, then it is rejected and simply a new xtriisgenerated in bounds. Each time the trial is accepted, xtrireplaces x (the centre x moves,but dx stays). At the end of the iteration, dx is scaled to some new length, which wouldhave produced a roughly equal number of rejected and accepted trials; for instance, dx isincreased if too many cases have been accepted.The temperature is reduced by a factor 0.6 after each 5 iterations. The process ishalted if the optimum found remains unchanged during several subsequent temperaturecycles.5Figure 4: Collimator distributions for the lattice shown in Fig. 1 ( Amax= 9.1).6

Numerical Optimization of Collimator Jaw Orientations and Locations in the LHC

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