Tag Archives: matrix solution techniques

Prettying Up the 3D Graph Output

Posted on May 26, 2016 by R.P. Churchill

Here I’ve prettied the graph up so the lines change color every 100 degrees F, and I’ve added a legend. I suppose I could choose consecutive colors with a bit more contrast.

Posted in Tools and methods | Tagged animation, canvas, matrix solution techniques, simulation, sim_presentation_series, thermodynamics | Leave a comment

Graphing the Matrix Output

Posted on May 25, 2016 by R.P. Churchill

Here I continually run the heating routine for a piece with an initial temperature of 70 degrees in a furnace set to 2300 degrees. The temperature scale, expressed vertically, runs from 0-2500 degrees. The bottom of the piece, which sits … Continue reading →

Posted in Tools and methods | Tagged animation, canvas, matrix solution techniques, simulation, sim_presentation_series, thermodynamics | Leave a comment

How Quickly Can the Matrix Be Solved On Different Machines?

Posted on May 24, 2016 by R.P. Churchill

Today I ran 2750 iterations of the matrix on some additional machines. My 64GB iPad 3 runs a 1 GHz, dual-core, 32-bit, ARM Cortex-A9 and ran the test in 1.308 seconds on load and about 1.15 seconds on rerun. My … Continue reading →

Posted in Tools and methods | Tagged changing technology, efficiency, JavaScript, matrix solution techniques, sim_presentation_series | Leave a comment

How Quickly Can the Matrix Be Solved?

Posted on May 23, 2016 by R.P. Churchill

The solution was finally made to run last week. Today the question is how fast the thing runs. My feel for the answer to this question has to do with the context in which I first asked it. From 1994 … Continue reading →

Posted in Tools and methods | Tagged changing technology, efficiency, JavaScript, matrix solution techniques, model-predictive control, simulation, sim_presentation_series | Leave a comment

Finally Solving the Matrix

Posted on May 19, 2016 by R.P. Churchill

Once we get all the parameters calculated and the matrix loaded, then we can finally solve the thing. Here’s how the code looks:

    function loadMatrix(index,fceTemp)
    {
      var i; var j; var k;
      var tsa; var tfa; var ts3; var tf4;

      tfa = arrayTfce[index] + Tabs;
      tf4 = tfa * tfa * tfa * tfa;

      for (k = 0; k <= sizeNodes; k++)
      {
        ZSRCE[k] = 0.0;
        ZAUX[k] = 0.0;
        for (i = 0; i <= sizeBandWidth; i++)
          ZMATX[k][i] = 0.0;
      }

      k = 0;
      for (j = 0; j < nodesHigh; j++)
        for (i = 0; i < nodesWide; i++)
        {
          k++;

          //nodal energy
          ZMATX[k][1] += arrayRho[i][j] * arrayCp[i][j];
          ZSRCE[k] += arrayRho[i][j] * arrayCp[i][j] * arrayT[i][j];

          //conduction to right
          if (arrayKright[i][j] > 0.0)
          {
            ZMATX[k][1] += arrayKright[i][j];
            ZMATX[k][2] -= arrayKright[i][j];
            ZMATX[k+1][1] += arrayKright[i][j];
          }

          //conduction up
          if (arrayKup[i][j] > 0.0)
          {
            ZMATX[k][1] += arrayKup[i][j];
            ZMATX[k][1+nodesWide] -= arrayKup[i][j];
            ZMATX[k+nodesWide][1] += arrayKup[i][j];
          }

          //radiation
          if (arrayRad[i][j] > 0.0)
          {
            tsa = arrayT[i][j];
            ts3 = tsa * tsa * tsa;
            ZMATX[k][1] += arrayRad[i][j] * ts3;
            ZSRCE[k] += arrayRad[i][j] * tf4;
          }

        }
    }     //loadMatrix

    function oneCalc(index) {
      calcParams(index);

      for (var i=0; i<nodesWide; i++) {
        for (var j=0; j<nodesHigh; j++) {
          arrayT[i][j] = arrayTpieces[index][i][j] + Tabs;
        }
      }
      
      loadMatrix(index);
      calcMatrix();
      
      k = 0;
      for (var j=0; j<nodesHigh; j++) {
        for (var i=0; i<nodesWide; i++) {
          k++;
          arrayTpieces[index][i][j] = ZSRCE[k] - Tabs;
        }
      }
    }  //oneCalc

    var loopCount = 0;
    
    var s="Temperatures after "+loopCount+" time steps<br />";
    for (var j=nodesHigh-1; j>=0; j--) {
      for (var i=0; i<=nodesWide-1; i++) {
        s += arrayTpieces[0][i][j].toPrecision(7) + ", ";
      }
      s += "<br />";
    }
    document.getElementById("stuff").innerHTML = s;

    function runOneTime() {
      if (loopCount == 0) {
        for (var i=0; i< pieceCount; i++) {
          for (var j=0; j<nodesWide; j++) {
            for (var k=0; k<nodesHigh; k++) {
              arrayTpieces[i][j][k] = 70.0;
            }
          }
        }
      } else {
        oneCalc(0);
      }
      var s="Temperatures after "+loopCount+" time steps<br />";
      for (var j=nodesHigh-1; j>=0; j--) {
        for (var i=0; i<=nodesWide-1; i++) {
          s += arrayTpieces[0][i][j].toPrecision(7) + ", ";
        }
        s += "<br />";
      }
      document.getElementById("stuff").innerHTML = s;
      loopCount++;
      if (loopCount > 20) {
        loopCount = 0;
      }
    }

    var intervalID = setInterval(runOneTime, 1000);

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function loadMatrix(index,fceTemp)

{

var i; var j; var k;

var tsa; var tfa; var ts3; var tf4;

tfa = arrayTfce[index] + Tabs;

tf4 = tfa * tfa * tfa * tfa;

for (k = 0; k <= sizeNodes; k++)

{

ZSRCE[k] = 0.0;

ZAUX[k] = 0.0;

for (i = 0; i <= sizeBandWidth; i++)

ZMATX[k][i] = 0.0;

}

k = 0;

for (j = 0; j < nodesHigh; j++)

for (i = 0; i < nodesWide; i++)

{

k++;

//nodal energy

ZMATX[k][1] += arrayRho[i][j] * arrayCp[i][j];

ZSRCE[k] += arrayRho[i][j] * arrayCp[i][j] * arrayT[i][j];

//conduction to right

if (arrayKright[i][j] > 0.0)

{

ZMATX[k][1] += arrayKright[i][j];

ZMATX[k][2] -= arrayKright[i][j];

ZMATX[k+1][1] += arrayKright[i][j];

}

//conduction up

if (arrayKup[i][j] > 0.0)

{

ZMATX[k][1] += arrayKup[i][j];

ZMATX[k][1+nodesWide] -= arrayKup[i][j];

ZMATX[k+nodesWide][1] += arrayKup[i][j];

}

//radiation

if (arrayRad[i][j] > 0.0)

{

tsa = arrayT[i][j];

ts3 = tsa * tsa * tsa;

ZMATX[k][1] += arrayRad[i][j] * ts3;

ZSRCE[k] += arrayRad[i][j] * tf4;

}

} //loadMatrix

function oneCalc(index) {

calcParams(index);

for (var i=0; i<nodesWide; i++) {

for (var j=0; j<nodesHigh; j++) {

arrayT[i][j] = arrayTpieces[index][i][j] + Tabs;

}

loadMatrix(index);

calcMatrix();

k = 0;

for (var j=0; j<nodesHigh; j++) {

for (var i=0; i<nodesWide; i++) {

k++;

arrayTpieces[index][i][j] = ZSRCE[k] - Tabs;

}

} //oneCalc

var loopCount = 0;

var s="Temperatures after "+loopCount+" time steps<br />";

for (var j=nodesHigh-1; j>=0; j--) {

for (var i=0; i<=nodesWide-1; i++) {

s += arrayTpieces[0][i][j].toPrecision(7) + ", ";

}

s += "<br />";

}

document.getElementById("stuff").innerHTML = s;

function runOneTime() {

if (loopCount == 0) {

for (var i=0; i< pieceCount; i++) {

for (var j=0; j<nodesWide; j++) {

for (var k=0; k<nodesHigh; k++) {

arrayTpieces[i][j][k] = 70.0;

}

} else {

oneCalc(0);

}

var s="Temperatures after "+loopCount+" time steps<br />";

for (var j=nodesHigh-1; j>=0; j--) {

for (var i=0; i<=nodesWide-1; i++) {

s += arrayTpieces[0][i][j].toPrecision(7) + ", ";

}

s += "<br />";

}

document.getElementById("stuff").innerHTML = s;

loopCount++;

if (loopCount > 20) {

loopCount = 0;

}

var intervalID = setInterval(runOneTime, 1000);

We start with a 7×7-node workpiece where the interior nodes are two inches wide and two … Continue reading →

Posted in Tools and methods | Tagged matrix solution techniques, sim_presentation_series, thermodynamics | Leave a comment

Dimensions and Other Considerations Before We Solve the Matrix

Posted on May 18, 2016 by R.P. Churchill

Yesterday’s post showed how some of the constants and parameters are initialized or calculated. Today we’ll describe a few more. This exercise defines a cross-section of a steel billet that is 8 inches by 8 inches and divided into five … Continue reading →

Posted in Tools and methods | Tagged matrix solution techniques, sim_presentation_series, thermodynamics | Leave a comment

Static and Dynamic Coefficients in Matrix Thermal Solutions

Posted on May 17, 2016 by R.P. Churchill

Recalling yesterday’s initial heat balance equation for each node: it should be noted that the values of both the specific heat and the thermal conductivity of steel are themselves functions of temperature. That means that these values need to be … Continue reading →

Posted in Tools and methods | Tagged matrix solution techniques, simulation, sim_presentation_series, thermodynamics | Leave a comment

A Specific Example of Equations To Be Solved By Matrix

Posted on May 16, 2016 by R.P. Churchill

Dividing a cross-section of a steel billet into nodes, we can start by looking at the energy balance for each node: All this equation is saying is that the change in the quantity of energy in a given node per … Continue reading →

Posted in Tools and methods | Tagged matrix solution techniques, sim_presentation_series, thermodynamics | Leave a comment

Implementing the Efficient Matrix Solution for Special Classes of Engineering Problems

Posted on May 12, 2016 by R.P. Churchill

Yesterday I described how to set up a matrix for efficient solution, today I’ll describe how to actually execute the solution. I plan to go back and revisit the intermediate steps but I wanted to show how this series of … Continue reading →

Posted in Software | Tagged efficiency, matrix solution techniques, sim_presentation_series, Software | Leave a comment

Efficient Matrix Solutions for Special Classes of Engineering Problems

Posted on May 11, 2016 by R.P. Churchill

There are a large class of analogous engineering problems that may be addressed with a particular mathematical solution. Fluid systems where volumes containing fluid are connected by fluid flows, mechanical systems where masses are connected by springs and dashpots (which … Continue reading →

Posted in Tools and methods | Tagged efficiency, engineering problems, Experience, matrix solution techniques, numerical methods, sim_presentation_series | Leave a comment

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