Modeling: Simulation Examples for Distributed Parameters Processes
Let us consider equations, or systems of equations with partial derivatives, used especially in engineering applications.
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Modeling: Simulation Examples for Distributed Parameters Processes
8.1 The Performance Indicator of Numerical Integration Let us consider equations, or systems of equations with partial derivatives, used especially in engineering applications. From the three types of solutions, general, particular or singular, the particular solutions, respectively, usually the exponential or polynomial ones can approximate quite well the numerous processes, from the control systems of electroenergetic, thermo-energetic, chemical and electromechanical processes. For the performance indicator of numerical integration, we will also consider ‘‘the cumulative relative error in percent’’ (crep) of the form: Pk f k jDxk j crep y ¼ crep x0...0 ¼ 100 Pkf 0 ð8:1Þ k0 jyANK j where yAN; k is the particular analytical solution (or in the optimum case, general) and Dxk ¼ x0...k yAN; k is the error of the solution numerically approximated (x0…k) compared with the analytical solution (yAN, k). Both components at the nominator and denominator of (8.1) are considered as having absolute values. The Pk notation ð kf0 Þ symbolizes the iterative sum of all sequences of calculus, from k0 ¼ t0 =Dt to kf ¼ tf =Dt, where (t0) and (tf) correspond to the initial and final moment, and (Dt) is the integration step, considered to be small enough. Example PDE I.2 is of the form: a00 y þ a10
oy oy þ a01 ¼ uðt; pÞ ot op
ð8:2Þ
or a00 x00 þ a10 x10 þ a01 x01 ¼ u00 :
T. Colosi et al., Numerical Simulation of Distributed Parameter Processes, DOI: 10.1007/978-3-319-00014-5_8, Springer International Publishing Switzerland 2013
ð8:3Þ
83
84
8 Modeling: Simulation Examples
We obtain x10 ¼
1 ½u ða00 x00 þ a01 x01 Þ a10 00
ð8:4Þ
from which the partial derivatives are calculated: x1þT; P ¼
1 ½u ða00 xTP þ a01 xT;1þP Þ a10 TP
ð8:5Þ
for P ¼ 1; 2; . . .30 and T ¼ 1; 2; . . .6. It results in the ‘‘matrix with partial derivatives of the state vector’’:
M pdx =
x xT
xP xTP
x 00 x10 = x 20
x 01 x11 x 21
x 02 x12 x 22
x 0,30 x1,30 x 2,30
x 60
x 61
x 62
x 6,30
ð8:6Þ
The initial conditions (IC) at t = t0 = 0 and k = k0 = 0 are considered to be known, xIC = x (t0, p), from which, by means of analytical derivative with respect to (p), we obtain all the elements of the line vector xP (t0, p). The transfer from the sequence (k - 1) to the sequence (k) is obtained: xk ffi xk1 þ
6 X DtT T¼1
T!
ð8:7Þ
xT; k1
and xP;k ffi xP; k1 þ
6 X DtT T¼1
T!
xTP;k1 :
ð8:8Þ
It was operated with the particular solution: yAN ðt; pÞ ¼ y00 þ ðJ0T þ J1T et=T1 þ J2T et=T2 ÞðJ0P þ J1P ep=P1 þ J2P ep=P2 Þ Ku u ð8:9Þ where : y00 ¼ 1; t0 ¼ 0; tf ¼ 10; p0 ¼ 0; pf ¼ 10; Ku ¼ 100; u ¼ 10; kt ¼ 1:5; tf pf lt ¼ 3; kp ¼ 2; lp ¼ 3; T1 ¼ l ð1þk ; T2 ¼ kt T1 ; P1 ¼ l ð1þk ; P2 ¼ kp P1 ; tÞ pÞ t
p
1 2 1 2 J0T ¼ 1; J1T ¼ T1TT ; J2T ¼ T2TT ; J0P ¼ 1; J1P ¼ P1PP ; J2P ¼ P2PP : 2 1 2 1 Using the method of calculus presented in (4.3), for the particular solution (8.9), a00 = 1, 0.1 B a10 B 10 and 0.1 B a01 B 10 with the integration step Dt
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