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Cumulative definte integral makes kernel run slow

Kaushik Mallick

Posted 10 years ago

I have a problem where I need to calculate the cumulative definite integral and mind the min and max of the function. I believe I have set it up correctly and I have been able to compute what I need for a simple case. But during the evaluation the Mathematica kernel runs for along time and locks up my PC. I would appreciate any help or suggestion you may have to improve the code. Clear["Global`*"]; t =.; f = 1; \[Omega] = 2 \[Pi] f; Tp = 1/f; r = 1.; L = 6 r; \[Theta] = \[Omega] t; Vd =.; Ncyl = 3; \[Phi][i_] := (i - 1) 2 \[Pi]/Ncyl; Q[t_, i_] := r Sin[\[Theta] + \[Phi][i]] - ( r^2 Sin[\[Theta] + \[Phi][i]] Cos[\[Theta] + \[Phi][i]])/Sqrt[ L^2 - r^2 (Sin[\[Theta] + \[Phi][i]])^2]; ftot[t_] := Sum[Max[Q[t, i], 0], {i, 1, Ncyl}]; favg = Quiet@NIntegrate[ftot[t], {t, 0, Tp}, AccuracyGoal -> 2]/Tp; Vstroke = favg Tp; fd[t_] := ftot[t] - favg; Plot[{ftot[t], favg, fd[t]}, {t, 0, Tp}, Frame -> True, Axes -> True, PlotRange -> { {0, Tp}, {-1.2, 1.2}}, FrameLabel -> {"Time (sec)", "Flow rate"}, GridLines -> Automatic, AspectRatio -> .65, PlotStyle -> {{Red, Thickness[0.005]}, {Green, Thickness[0.0051]}, {Blue, Thickness[0.005]}}, BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, FontFamily -> "Arial"}] Vd[s_] := Integrate[fd[t], {t, 0, s}]; Plot[{Vd[s]}, {s, 0, Tp}, Frame -> True, Axes -> True, FrameLabel -> {"Time (sec)", "Flow"}, GridLines -> Automatic, AspectRatio -> .65, PlotStyle -> {{Brown, Thickness[0.005]}, {Cyan, Thickness[0.0051]}, {Gray, Thickness[0.01]}}, BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, FontFamily -> "Arial"}] Vmax = MaxValue[Vd[s], s]; Vmin = MinValue[Vd[s], s]; deltav = (Vmax - Vmin)/Vstroke

I have a problem where I need to calculate the cumulative definite integral and mind the min and max of the function. I believe I have set it up correctly and I have been able to compute what I need for a simple case. But during the evaluation the Mathematica kernel runs for along time and locks up my PC. I would appreciate any help or suggestion you may have to improve the code.

Clear["Global`*"];
t =.; f = 1; \[Omega] = 2 \[Pi] f; Tp = 1/f;
r = 1.;
L = 6 r;
\[Theta] = \[Omega] t;
Vd =.;
Ncyl = 3;
\[Phi][i_] := (i - 1) 2 \[Pi]/Ncyl;
Q[t_, i_] := 
  r Sin[\[Theta] + \[Phi][i]] - (
   r^2 Sin[\[Theta] + \[Phi][i]] Cos[\[Theta] + \[Phi][i]])/Sqrt[
   L^2 - r^2 (Sin[\[Theta] + \[Phi][i]])^2];

ftot[t_] := Sum[Max[Q[t, i], 0], {i, 1, Ncyl}];

favg = Quiet@NIntegrate[ftot[t], {t, 0, Tp}, AccuracyGoal -> 2]/Tp;
Vstroke = favg Tp;
fd[t_] := ftot[t] - favg;

Plot[{ftot[t], favg, fd[t]}, {t, 0, Tp}, Frame -> True, Axes -> True, 
 PlotRange -> { {0, Tp}, {-1.2, 1.2}}, 
 FrameLabel -> {"Time (sec)", "Flow rate"}, GridLines -> Automatic, 
 AspectRatio -> .65, 
 PlotStyle -> {{Red, Thickness[0.005]}, {Green, 
    Thickness[0.0051]}, {Blue, Thickness[0.005]}}, 
 BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, 
   FontFamily -> "Arial"}]

Vd[s_] := Integrate[fd[t], {t, 0, s}];
Plot[{Vd[s]}, {s, 0, Tp}, Frame -> True, Axes -> True, 
 FrameLabel -> {"Time (sec)", "Flow"}, GridLines -> Automatic, 
 AspectRatio -> .65, 
 PlotStyle -> {{Brown, Thickness[0.005]}, {Cyan, 
    Thickness[0.0051]}, {Gray, Thickness[0.01]}}, 
 BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, 
   FontFamily -> "Arial"}]

Vmax = MaxValue[Vd[s], s];
Vmin = MinValue[Vd[s], s];
deltav = (Vmax - Vmin)/Vstroke

POSTED BY: Kaushik Mallick

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Kaushik Mallick

Posted 10 years ago

That did it! Thanks a lot.

POSTED BY: Kaushik Mallick

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 10 years ago

Sorry, that should probably be NMaxValue[ {vd[s], 0<=s<=Tp}, s]

POSTED BY: Gianluca Gorni

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 10 years ago

I think you should add the domain constraints: NMaxValue[{vd,0<=s<=Tp}, s]

POSTED BY: Gianluca Gorni

Kaushik Mallick

Posted 10 years ago

Gianluca, That was an excellent idea. Thanks for your suggestion. I had to make minor adjustment to the code, but it made it work. vd = NDSolveValue[{vd[0] == 0, vd'[s] == fd[t = s]}, vd, {s, 0, Tp}, AccuracyGoal -> 8]; However, I am getting some errors associated with computing the min and max value of the function. Any suggestion on what I need to do to avoid these errors? Clear["Global`*"]; t =.; f = 1; \[Omega] = 2 \[Pi] f; Tp = 1/f; r = 1.; L = 6 r; \[Theta] = \[Omega] t; Vd =.; Ncyl = 3; \[Phi][i_] := (i - 1) 2 \[Pi]/Ncyl; Q[t_, i_] := r Sin[\[Theta] + \[Phi][i]] - ( r^2 Sin[\[Theta] + \[Phi][i]] Cos[\[Theta] + \[Phi][i]])/Sqrt[ L^2 - r^2 (Sin[\[Theta] + \[Phi][i]])^2]; ftot[t_] := Sum[Max[Q[t, i], 0], {i, 1, Ncyl}]; favg = Quiet@NIntegrate[ftot[t], {t, 0, Tp}, AccuracyGoal -> 4]/Tp; vstroke = favg Tp; fd[t_] := ftot[t] - favg; Plot[{ftot[t], favg, fd[t]}, {t, 0, Tp}, Frame -> True, Axes -> True, PlotRange -> { {0, Tp}, {-1.2, 1.2}}, FrameLabel -> {"Time (sec)", "Flow rate"}, GridLines -> Automatic, AspectRatio -> .65, PlotStyle -> {{Red, Thickness[0.005]}, {Green, Thickness[0.0051]}, {Blue, Thickness[0.005]}}, BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, FontFamily -> "Arial"}] vd = NDSolveValue[{vd[0] == 0, vd'[s] == fd[t = s]}, vd, {s, 0, Tp}, AccuracyGoal -> 8]; Plot[{vd[s]/vstroke}, {s, 0, Tp}, Frame -> True, Axes -> True, FrameLabel -> {"Time (sec)", "Flow"}, GridLines -> Automatic, AspectRatio -> .65, PlotStyle -> {{Brown, Thickness[0.005]}, {Cyan, Thickness[0.0051]}, {Gray, Thickness[0.01]}}, BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, FontFamily -> "Arial"}] vmax = NMaxValue[vd, s]; vmin = NMinValue[vd, s]; deltav = (vmax - vmin)/vstroke

Gianluca, That was an excellent idea. Thanks for your suggestion. I had to make minor adjustment to the code, but it made it work.

vd = NDSolveValue[{vd[0] == 0, vd'[s] == fd[t = s]}, vd, {s, 0, Tp}, 
   AccuracyGoal -> 8];

However, I am getting some errors associated with computing the min and max value of the function. Any suggestion on what I need to do to avoid these errors?

Clear["Global`*"];
t =.; f = 1; \[Omega] = 2 \[Pi] f; Tp = 1/f;
r = 1.;
L = 6 r;
\[Theta] = \[Omega] t;
Vd =.;
Ncyl = 3;
\[Phi][i_] := (i - 1) 2 \[Pi]/Ncyl;
Q[t_, i_] := 
  r Sin[\[Theta] + \[Phi][i]] - (
   r^2 Sin[\[Theta] + \[Phi][i]] Cos[\[Theta] + \[Phi][i]])/Sqrt[
   L^2 - r^2 (Sin[\[Theta] + \[Phi][i]])^2];

ftot[t_] := Sum[Max[Q[t, i], 0], {i, 1, Ncyl}];

favg = Quiet@NIntegrate[ftot[t], {t, 0, Tp}, AccuracyGoal -> 4]/Tp;
vstroke = favg Tp;
fd[t_] := ftot[t] - favg;

Plot[{ftot[t], favg, fd[t]}, {t, 0, Tp}, Frame -> True, Axes -> True, 
 PlotRange -> { {0, Tp}, {-1.2, 1.2}}, 
 FrameLabel -> {"Time (sec)", "Flow rate"}, GridLines -> Automatic, 
 AspectRatio -> .65, 
 PlotStyle -> {{Red, Thickness[0.005]}, {Green, 
    Thickness[0.0051]}, {Blue, Thickness[0.005]}}, 
 BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, 
   FontFamily -> "Arial"}]

vd = NDSolveValue[{vd[0] == 0, vd'[s] == fd[t = s]}, vd, {s, 0, Tp}, 
   AccuracyGoal -> 8];


Plot[{vd[s]/vstroke}, {s, 0, Tp}, Frame -> True, Axes -> True, 
 FrameLabel -> {"Time (sec)", "Flow"}, GridLines -> Automatic, 
 AspectRatio -> .65, 
 PlotStyle -> {{Brown, Thickness[0.005]}, {Cyan, 
    Thickness[0.0051]}, {Gray, Thickness[0.01]}}, 
 BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, 
   FontFamily -> "Arial"}]

vmax = NMaxValue[vd, s];
vmin = NMinValue[vd, s];
deltav = (vmax - vmin)/vstroke

POSTED BY: Kaushik Mallick

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 10 years ago

For cumulative integrals it may be convenient to use NDSolve instad of Integrate, at least if the interval is not too large: Vd = NDSolveValue[{Vd[0] == 0, Vd'[s] == fd[s]}, Vd, {s, 0, Tp}];

POSTED BY: Gianluca Gorni

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