1. 1 A Core Course on Modeling Enter: ACCEL basics running a script Time in ACCEL the { … } operator examples Week 3 – Time for Change

2. 2 A Core Course on Modeling ACCEL Week 3 – Time for Change intro and PR line-by-line, navigate a script assistance enter a script as text numerical analysis optimization larger view on simulation output

3. 3 A Core Course on Modeling ACCEL Week 3 – Time for Change p=5 q=slider(10,0,20) r=p+q Example of a script: Every line introduces a quantity Quantities can be constants (p) Quantities can be functions r = f(p,q) Quantities can be user-entered (q) paste into IO/edit box click ‘run’

4. 4 A Core Course on Modeling ACCEL Week 3 – Time for Change If something STRANGE happens: don't panic goto IO/edit ctrl-A (select all) ctrl-C (copy all) ctrl-V into text editor to save your script reload ACCEL goto IO/edit ctrl-V to load script into ACCEL retry image: http://partlycloudyjuly.wordpress.com/2011/02/25/look-both-ways/

5. 5 A Core Course on Modeling ACCEL Week 3 – Time for Change s=slider(10,0,20) // this is a slider r=p+q // this is an expression p=5 // this is a constant q=s+t // this is an expression t=pow(s,3) // this is a standard // function Example of a script with comment: to see values: click ‘show/hide values’ to see dependencies: click on ‘pauze’, next click on any quantity

6. 6 A Core Course on Modeling ACCEL Week 3 – Time for Change s=slider(10,0,100) z=plot([gr1,gr2]) gr1=[str,[s]] gr2=[str,[s % 10]] str='x:{mode:intp},y:{mode:shift,ref:1}' Example of a script with visual output: plot([graph 1,graph 2,…, graph n ]) plots n graphs graph i = [format,data] format = string, e.g. ' x:{mode:intp},y:{mode:shift,ref:1}' data = one or more quantities Black Magic for now …

7. 7 A Core Course on Modeling time in ACCEL Week 3 – Time for Change Remember recursive functions: Q current = f (Q prev, P prev ) Simplest example: time current = time prev + 1

8. 8 A Core Course on Modeling time in ACCEL Week 3 – Time for Change p=t t=t{1}+1 ACCEL uses {1} to access previous value ACCEL uses {n} to access n-th previous value, n>0 ACCEL initializes after modification in script Start conditions: ACCEL initializes all quantities to 0, unless you use { … |(value in case historic value is lacking)}

9. 9 A Core Course on Modeling time in ACCEL Week 3 – Time for Change p=k k=if(t>0,k{1}+5,0) t=t{1}+1 ACCEL can be forced to re-initialize after nr steps:

10. 10 A Core Course on Modeling time in ACCEL Week 3 – Time for Change z=descartes([gr1,gr2]) gr1=[str1,[s]] gr2=[str2,[50+s-s{1}]] s=50+25*sin(t/10) z=descartes([gr1,gr2]) gr1= [locations:[x:[mode:'intp'],y:[mode:'shift',value:s]],edges:[thickness:2,col_r:255]] gr2= [locations:[x:[mode:'intp'],y:[mode:'shift',value:50+s- s{1}]],edges:[thickness:2,col_b:255]] t=t{1}+1 Implement first derivative image: http://www.treklens.com/gallery/photo593123.htm

11. 11 A Core Course on Modeling time in ACCEL Week 3 – Time for Change z=descartes([gr1,gr2]) gr1= [locations:[x:[mode:'intp'],y:[mode:'shift',value:50+10*rate]],edges:[thickness:2,co l_r:if(rate>0,0,255),col_g:if(rate>0,255,0)]] gr2= [locations:[x:[mode:'intp'],y:[mode:'shift',value:50+supply]],edges:[thickness:2,col _b:255]] rate=slider(0,-0.5,0.5) supply=supply{1}+rate Implement integral image: http://www.themarketingexpert.net/2011/12/leaky-bucket.html

12. 12 A Core Course on Modeling time in ACCEL Week 3 – Time for Change z=descartes([gr2,gr1]) gr1= [locations:[x:[mode:'intp'],y:[mode:'shift',value:s]],edges:[thickness:2, col_r:255]] gr2= [locations:[x:[mode:'intp'],y:[mode:'shift',value:ds]],edges:[thickness:2,col_b:255]] s=slider(50,0,100) ds=(1-damp)*s+damp*ds{1} damp=slider(0.5,0.01,0.99) Implement damping image: http://www.acecontrols.co.uk/product-range

13. 13 A Core Course on Modeling time in ACCEL Week 3 – Time for Change z=descartes([gr1,gr2]) gr1= [locations:[x:[mode:'intp'],y:[mode:'shift',value: s ]],edges:[th ickness:2,col_r:255]] gr2= [locations:[x:[mode:'intp'],y:[mode:'shift',value: s{delay} ] ],edges:[thickness:2,col_b:255]] s=slider(50,0,100) delay=slider(1,1,100) Implement delay image: http://www.rainbowresource.com/prodlist.php?subject=20&category=8753

14. t  i  K(t)  K i u(t)  u i u’(t)  (u(t+  )-u(t))/  = (u i+1 -u i )/  v’(t)  (v(t+  )-v(t))/  = (v i+1 -v i )/  So u i+1 = u i +  v i v i+1 = v i +  a i = v i+1 = v i +  a i = = v i +  K i /m = = v i +  K i /m = = v i +  C(u rest -u i ) /m = v i +  C(u rest -u i ) /m Recursive function Q curr =F(Q prev,P prev ): u curr =F 1 (u prev,v prev )=u prev +  v prev v curr =F 2 (v prev,u prev )= =v prev +  C(u rest -u prev ) /m, =v prev +  C(u rest -u prev ) /m, with suitable u 0 and v 0 Equal intervals : a mass-spring system with damping. From physics, we know K=ma where K=C(u rest -u) and a=u’’. Write v=u’, then a=v’. Here, K=K(t), u=u(t). Let us approximate by sampling. For   0: out 14 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

15. t  i  K(t)  K i u(t)  u i u’(t)  (u(t+  )-u(t))/  = (u i+1 -u i )/  v’(t)  (v(t+  )-v(t))/  = (v i+1 -v i )/  So u i+1 = u i +  v i v i+1 = v i +  a i = v i+1 = v i +  a i = = v i +  K i /m = = v i +  K i /m = = v i +  C(u rest -u i ) /m = v i +  C(u rest -u i ) /m Recursive function Q curr =F(Q prev,P prev ): u curr =F 1 (u prev,v prev )=u prev +  v prev v curr =F 2 (v prev,u prev )= =v prev +  C(u rest -u prev ) /m, =v prev +  C(u rest -u prev ) /m, with suitable u 0 and v 0 Equal intervals : a mass-spring system with damping. From physics, we know K=ma where K=C(u rest -u) and a=u’’. Write v=u’, then a=v’. Here, K=K(t), u=u(t). Let us approximate by sampling. For   0: -v-v-v-v ( -  v prev ) ( -  v i ) out 15 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

16. Equal intervals WARNING:   0 is not the same as '  =small'.   0 is not the same as '  =small'. The substitutions for u’(t) and v’(t) are approximations only. As we will see in a later example, aliasing errors are worse when the sampled signal changes rapidly, compared to the sampling rate. Concrete: unless  is much smaller than the period of the oscillation (=  m/C), the approximations are bad – with the risk of instability. 16 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

17. out Infinitesimal intervals the mass-spring system with damping revisited. From physics, we know K=ma where K=C(u rest -u) and a=u’’. For some purposes the numerical solution (sampling) is not acceptable. Interested in outcome?  use better numerical methods Interested in insight?  try to use symbolic methods, i.e.: analysing differential equations Only possible in very few special cases In particular: linear (sets of) DE’s. 17 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

18. Infinitesimal intervals the mass-spring system with damping revisited. From physics, we know K=ma where K=C(u rest -u) and a=u’’. Here, K=K(t), u=u(t). Let us try a solution of the form u=u rest +A sin(t/T) u’’= -AT -2 sin(t/T) Substitute back: -mAT -2 sin(t/T)=C(u rest -u rest -Asin(t/T) ), mT -2 =C, or T=  m/C ( =:T 0 ) out 18 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

19. Infinitesimal intervals the mass-spring system with damping revisited. From physics, we know K=ma where K=C(u rest -u) and a=u’’. Here, K=K(t), u=u(t). try u=u rest +Ae t, then C+  +m 2 =0, or 1,2 =(-  (  2 -4mC))/2m 1,2 =(-  (  2 -4mC))/2m Let  0 =  (4mC). For  =  0  critical damping; for  <  0  oscillations; T=T 0 /  (1- (  /  0 ) 2 ) for  >  0  super critical damping (further details: lectures ‘dynamical systems’) out -v-v-v-v By substituting u = Ae t into C(u rest -u)-  u’=mu’’ and using u’=A e t ; u’’=A 2 e t. C(u rest -u rest -Ae t )-  A e t =mA 2 e t Must hold for any t, so indeed C +  + m 2 =0 19 A Core Course on Modeling Week 3 – Time for Change time in ACCEL

20. 20 A Core Course on Modeling time in ACCEL Week 3 – Time for Change c=slider(5,1,20.0) delta=slider(0.1,0.001,0.2) m=slider(5,1,10) mu=slider(0,0,1.5) reset=button() z=descartes([gr1,gr2]) gr1=[locations:[x:[mode:'intp'],y:[mode:'shift',value:40+10*u]],edges:[thickness:2,col_r:255]] gr2=[locations:[x:[mode:'intp'],y:[mode:'shift',value:45+10*u_exact]],edges:[thickness:2,col_g:255]] F=-c*u{1}-mu*v{1} a=F/m discr=mu*mu-4*m*c lambda_img=cond(discr>0,0,sqrt(-discr)/(2*m)) lambda_real=cond(discr>0,(-mu+sqrt(discr))/(2*m),-mu/(2*m)) u_exact=cond(discr>0,exp(lambda_real*t),exp(lambda_real*t)*cos(lambda_im g*t)) t=if(reset,0,t{1}+delta) u=if(reset,1,u{1}+delta*v) v=if(reset,0,v{1}+delta*a) A mass-spring system user input graphics force and acceleration exactsolution recursive functions

21. 21 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A cannon shot simulation http://thehumanmarvels.com/1634/zazel-the-human-cannonball/talents x y  y(t) = v y t + gt 2 /2 x(t) = v x t y(t) = 0, so t = 2 v y / g Then x = 2 v x v y / g  2 (cos  sin  ) / g = sin 2  /g Maximize x:  = 45 o But what in case of damping?

22. 22 A Core Course on Modeling time in ACCEL Week 3 – Time for Change airDamp=slider(0.005,0,0.03) dt=slider(0.1,0.02,40.0) phi=slider(0,0,1.5) plotBall=descartes([[ locations:[fill:'interior',x:r.x,y:r.y,rad:2.5 ]]])px=r.x g=-0.003 groundDamp=0.02 v=0.55 boom=(r{1}.vx<0.000001) newVx=r{1}.vx*(1-velDamp*dt) newVy=r{1}.vy+(g-velDamp*r{1}.vy)*dt newX=min(r{1}.x+dt*r{1}.vx,100) newY=max(r{1}.y+dt*r{1}.vy,0) r=if(boom,startR,simR) simR=['x':newX,'y':newY,'vx':newVx,'vy':newVy] startR=['x':0,'y':20,'vx':v*cos(phi),'vy':v*sin(phi)] velDamp=if(r{1}.y>0,airDamp,groundDamp) A cannon shot simulation

23. 23 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A combat simulation Lanchester's Law: e 1 : nr troops of army 1 e 2 : nr troops of army 2 1 : relative strength of 1 2 : relative strength of 2 then de 1 /dt = - 2 e 2 de 2 /dt = - 1 e 1 Given e 1 (t=0)=e 10, e 2 (t=0)=e 20, who wins? http://www.napolun.com/mirror/napoleonistyka.atspace.com/default.htm

24. 24 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A combat simulation p=check(true) e1=if(p,e1_0,max(0,e1{1}-lambda_2*e2{1})) e2=if(p,e2_0,max(0,e2{1}-lambda_1*e1{1})) lambda_1=slider(0.01,0,0.02) lambda_2=slider(0.01,0,0.02) e1_0=slider(100,0,500) e2_0=slider(100,0,500) res=descartes([graph1,graph2]) graph1=[ locations:[x:[mode:'intp'],y:[mode:'shift',value:e1/5],rad:1,fcol_r:255,fill:'interior'] ] graph2=[ locations:[x:[mode:'intp'],y:[mode:'shift',value:e2/5],rad:1,fcol_g:255,fill:'interior'] ]

25. 25 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A predator - prey simulation http://q-rai.deviantart.com/ Lotka – Volterra model: br = birth rate of rabit (autonomous) bf = birth rate of fox, feeding on rabit dr = rabits killed by fox df = death rate of fox (autonomous)  r /  t = r (br – dr f)  f /  t = f (bf r – df )

26. 26 A Core Course on Modeling time in ACCEL Week 3 – Time for Change bf=slider(1.5,0,5) br=slider(3,0,5) df=slider(1,0,5) dr=slider(1,0,5) dt=slider(0.15,0.01,0.2) reset=button() p=descartes([gboth,gfox,grab]) gboth=[ locations:[fill:'interior',nrLocations:50,rad:2.5,fcol_r:50,x:[mode:'shift',value:10+3*f ],y:[mode:'shift',value:10+3*r]] ] gfox=[ locations:[y:[mode:'intp'],x:[mode:'shift',10+3*f]],edges:[thickness:2,col_r:255] ] grab=[ locations:[x:[mode:'intp'],y:[mode:'shift',10+3*r]],edges:[thickness:2,col_b:255] ] f=if((time<2)||reset,150,max(0.01,f{1}+dt*(bf*f{1}*r{1}-df*f{1}))) r=if((time<2)||reset,150,max(0.01,r{1}+dt*(br*r{1}-dr*f{1}*r{1}))) time=time{1}+1 A predator - prey simulation user input graphics recursive functions time

27. 27 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A billiards simulation http://joyreactor.com/tag/billiards vrvr v' r vwvw v' w rr  w = -  r =  =:   = (v w -v r,  )  ; ||  ||=1

28. 28 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A billiards simulation vrvr v' r vwvw v' w rr  w = -  r =  =:   = (v w -v r,  )  ; ||  ||=1 Proof: Energy conservation: v r 2 +v w 2 =(v r +  ) 2 +(v w -  ) 2, hence v r 2 +v w 2 =(v r +  ) 2 +(v w -  ) 2, hence 0=(2  (v r,  )+  2 )+(-2  (v w,  )+  2 ), hence  = (v w -v r,  ), QED

29. 29 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A billiards simulation vrvr v' r vwvw v' w rr  w = -  r =  =:   = (v w -v r,  )  ; ||  ||=1 v' r =v r +(v w -v r,  )  v' w =v w +(v r -v w,  )  Proof: Energy conservation: v r 2 +v w 2 =(v r +  ) 2 +(v w -  ) 2, hence v r 2 +v w 2 =(v r +  ) 2 +(v w -  ) 2, hence 0=(2  (v r,  )+  2 )+(-2  (v w,  )+  2 ), hence  = (v w -v r,  ), QED

30. 30 A Core Course on Modeling time in ACCEL Week 3 – Time for Change A billiards simulation maxX=40 minX=-40 maxY=25 minY=-25 halfWay=50 // half of the screen; used to correct the queueX and queueY coordinates help=check(false) // plot the direction of the queue nBalls=3 // nr balls // This is also hard coded in the colors; if the nr balls should change, only the color // arrays have to be updated rhoBall=slider(1.4,0.6,4) // radius of the balls mBall=5 // mass of the balls time=time{1}+1 rollDamp=0.994 // the damping coefficient for the rolling balls collDamp=0.6 // the damping coefficient for colliding balls eThreshold=0.5 // the minimum kinetic energy for the balls to stay rolling dT=0.8 // to go from physical time to simulation time PR=plot([plotTable,plotShadow,plotBalls,plotHighLight,plotQueue,plotQueue2]) // one graph consisting of three balls. The x and y coordinates are // taken from the 'x'- and 'y'-components of the r-vector that has to be transposed in order to get // the x-s and y-s in two separate vectors. tVec(i,j)=@(r{1},j)-@(r{1},i) // the vector pointing from ball i to ball j, with coefficients x and y cpl=#(i,vSequence(0,nBalls),cplOneBall(i),vAppend) // the couplings matrix containing all info about the relations to ball i, // that is: 'close' to indicate if these two balls are in a collision-state; // if close is true, 'force' is the current reaction force between them. tMat=#(i,vSequence(0,nBalls),tMatOneBall(i),vAppend) // the tMatrix contains, for every pair, i-j, the t-vector between the centres tMatOneBall(i)=#(j,vSequence(0,nBalls),tVec(i,j),vAppend) // the tMatOneBall vector contains, for every ball, // the the t-vectors between its centre and the other centres touch(i,j)=((vNormEuclid(@(@(tMat,i),j))<(@(rho,i)+@(rho,j))) && (i!=j) && vDot(@(@(tMat,i),j),@(p{1},i)/@(m,i)-@(p{1},j)/@(m,j))>0) // condition for colliding contact. Three terms: // 1. Are the balls close enough? // 2. No self-collision? // 3. Is the relative velocity opposite to the vector connecting the centres? cplOneBall(i)=#(j,vSequence(0,nBalls),cond(touch(i,j),['close':true,'force':(mpt(i,j)/m1m2tt(i,j))*@(@(tMat, i),j)],['close':false,'force':0]),vAppend) // the cplOneBall vector contains, for ball i, the info for the collisions between this // ball and all other balls. // It sets the value to the boolean 'close', and if close==true, the // current force vector. The derivation of the force vector // is based on conservation of momentum, angular momentum and kinetic energy // in a coordinate-free version. mpt(i,j)=-2*vDot(@(m,j)*@(p{1},i)-@(m,i)*@(p{1},j),@(@(tMat,i),j)) // the product (((m1p2-m2p1)t),t): the numerator of the force vector m1m2tt(i,j)=(@(m,i)+@(m,j))*vDot(@(@(tMat,i),j),@(@(tMat,i),j)) // the product (m1+m2)(t,t): the denominator of the force vector f=#(i,vSequence(0,nBalls),forceOnOneBall(i),vAppend) // calculate the forces for all balls by concatenating forceOnOneBall(i)=#(j,vSequence(0,nBalls),cond(@(@(@(cpl,i),j),'close'),@(@(@(cpl,i),j),'force'),0),add) // adding the forces due to all other of balls rho=#(i,vSequence(0,nBalls),rhoBall,vAppend) // the radii of the balls m=#(i,vSequence(0,nBalls),mBall,vAppend) // the mass of the balls col=['red':[255,255,255],'grn':[0,255,255],'blu':[0,255,0]] // colors of the balls: first one is red, two is white, three is yellowish r=if(gameState=='roll',r{1}+dT*p/m,cond(time==1,#(i,vSequence(0,nBalls),['x':halfWay*(random()- 0.5),'y':0.5*halfWay*(random()-0.5)],vAppend),r{1})) // r is obtained by integrating v; at the starting time (==1) give the initial random positions p=if(gameState=='roll',rollDamp*wallCollide(p{1}+f),[['x':0,'y':0],['x':@(@(r{1},1),'x')- queueX,'y':@(@(r{1},1),'y')-queueY],['x':0,'y':0]]) // p is obtained by tame integrating f; at the starting time give the initial momenta. The only // non-vanishing initial momentum is the momentum of ball 1; this is obtained from the queue position minus the centre of ball 1. wallCollide(a)=wallCollideLeft(wallCollideRight(wallCollideBottom(wallCollideTop(a)))) // the collisions with all walls wallCollideLeft(a)=#(i,vDom(a),cond((@(@(r{1},i),'x')>(minX+@(rho,i))) || (@(@(a,i),'x')>0),@(a,i),['x':- collDamp*@(@(a,i),'x'),'y':collDamp*@(@(a,i),'y')]),vAppend) // collide left wall wallCollideRight(a)=#(i,vDom(a),cond((@(@(r{1},i),'x')<(maxX-@(rho,i))) || (@(@(a,i),'x')<0),@(a,i),['x':- collDamp*@(@(a,i),'x'),'y':collDamp*@(@(a,i),'y')]),vAppend) // collide right wall wallCollideBottom(a)=#(i,vDom(a),cond((@(@(r{1},i),'y')>(minY+@(rho,i))) || (@(@(a,i),'y')>0),@(a,i),['x':collDamp*@(@(a,i),'x'),'y':-collDamp*@(@(a,i),'y')]),vAppend) // collide bottom wall wallCollideTop(a)=#(i,vDom(a),cond((@(@(r{1},i),'y')<(maxY-@(rho,i))) || (@(@(a,i),'y')<0),@(a,i),['x':collDamp*@(@(a,i),'x'),'y':-collDamp*@(@(a,i),'y')]),vAppend) // collide top wall eKin=#(i,vSequence(0,nBalls),vDot(@(p,i),@(p,i))/(2*@(m,i)),add) // the kinetic energy gameState=cond((eKin{1}<eThreshold) && !queueHit,'hit',cond(queueHit,'roll','hit')) // determine which state we are in queueHit=cursorB() cX=cursorX() cY=cursorY() queueX=cX-halfWay queueY=cY-halfWay // the queue end position is derived from the location of the cursor plotQueue=cond(gameState=='hit',['plotType:line,col_b:{value:0},col_g:{value:0},col_r:{value:0},x:{mode:data,ref:1},y:{mode:data,ref:2}',halfWay+[queueX,@(@(r{1},1),'x')],halfWay+[queueY,@(@(r{1},1),'y')]],['plo tType:line,x:{mode:data,ref:1},y:{mode:data,ref:2}',[0,0],[0,0]]) plotQueue2=cond(gameState=='hit' && help,['plotType:line,col_g:{value:100},col_r:{value:0},width:{value:0.4},x:{mode:data,ref:1},y:{mode:da ta,ref:2}',halfWay+[10*(@(@(r{1},1),'x')- queueX)+queueX,@(@(r{1},1),'x')],halfWay+[10*(@(@(r{1},1),'y')- queueY)+queueY,@(@(r{1},1),'y')]],['plotType:line,x:{mode:data,ref:1},y:{mode:data,ref:2}',[0,0],[0,0]] ) // the queue is drawn when in hit-mode plotTable=['plotType:vbar,yBase:{value:25},width:{value:80},height:{value:50},col_r:{value:0}',[]] // plot the table plotBalls=['plotType:bubble,x:{mode:data,ref:1},y:{mode:data,ref:2},diameter:{mode:data,ref:3},col_r:{mode:d ata,ref:4},col_g:{mode:data,ref:5},col_b:{mode:data,ref:6}',halfWay+@(vTranspose(r),'x'),halfWay+@(vTra nspose(r),'y'),2*rho,@(col,'red'),@(col,'grn'),@(col,'blu')] // plot the balls plotShadow=['plotType:bubble,x:{mode:data,ref:1},y:{mode:data,ref:2},diameter:{value:'+2*rhoBall+'},col_r:{v alue:0},col_g:{value:0},col_b:{value:0},col_a:{value:0.5}',halfWay- 0.15*rhoBall+@(vTranspose(r),'x'),halfWay-0.5*rhoBall+@(vTranspose(r),'y')] // plot the shadows beneath the balls plotHighLight=['plotType:bubble,x:{mode:data,ref:1},y:{mode:data,ref:2},diameter:{value:'+0.75*rhoBall+'},co l_r:{value:255},col_g:{value:255},col_b:{value:255}',halfWay+0.1*rhoBall+@(vTranspose(r),'x'),halfWay+0.5*rhoBall+@(vTranspose(r),'y')] // plot the highlights on the balls … and a further handfull of details