Code added

Author: Benjamin Horowitz

heathoundary.pde

@@ -0,0 +1,103 @@
+//Heat Differential Equations Visualization
+//Currently shows thermalization of a block with originally randomly generated internal temperatures. Can be later altered to follow a given distribution.
+
+// Ben Horowitz ([email protected])
+
+int bh = 300;
+int bw = 300;
+int[][] temp = new int[bh][bw];
+int[][] temp1 = new int[bh][bw];
+int mint=0;
+int maxt=100;
+float k = 0.5;
+
+
+ 
+void setup()
+{
+  size(600,600);
+  fill(255);
+  frameRate(80);
+  noStroke();
+  colorMode(HSB, 255);
+  intcond();
+}
+
+int t;
+
+
+  
+
+void intcond()
+{
+  
+  
+ float h;
+  h= height/bh;
+  float w;
+  w= width/bw;
+  
+ for(int i=0; i < bw; i+=1)
+  {
+      for(int j=0; j < bh; j+=1)
+      {
+       
+        
+       if(i==0 || i==bw-1 || j==0 ||  j==bh-1)
+      {
+        temp[i][j]=10;
+        temp1[i][j]=10;
+      }
+      else
+      {
+       temp[i][j] = int(random(256));
+       temp1[i][j]= temp[i][j];
+      }
+      }
+  }
+}
+
+void draw() 
+{ 
+  int[][] adjust;
+ float h;
+  h= height/bh;
+  float w;
+  w= width/bw;
+  t=t+1;
+  
+  for(int i=0; i*w < width; i+=1)
+  {
+    for(int j=0; j*h < height; j+=1)
+    {
+      int x;
+      int a;
+     
+      if(i==0 || i==bw-1 || j==0 ||  j==bh-1)
+      {
+        a=temp[i][j];
+      }
+      else
+      {
+      a= (temp1[i+1][j]+temp1[i-1][j]+temp1[i][j+1]+temp1[i][j-1])/4;
+      }
+      
+      x = temp[i][j] - a;
+      temp[i][j] = int(temp[i][j] - k*x);
+      float value = temp[i][j];  
+      fill(value, 255, 255);
+      rect(i*w,j*h,w,h);
+      
+    }
+    
+  }
+
+   for(int i=0; i*w < width; i+=1)
+  {
+    for(int j=0; j*h < height; j+=1)
+    {
+      temp1[i][j]=temp[i][j];
+    }
+  }
+} 
+

nbody.pde

@@ -0,0 +1,111 @@
+// N-Body Simulation including gravity and "sticky" collisions
+
+// Ben Horowitz ([email protected])
+
+int bh = 100;
+int bw = 100;
+int n = 60;
+float a = 0.0000000002*10;
+float[] pos_x = new float[n];
+float[] pos_y = new float[n];
+float[] vel_x = new float[n];
+float[] vel_y = new float[n];
+float[] mass = new float[n];
+float k = 0.5;
+
+
+ 
+void setup()
+{
+  size(700,700);
+  fill(255);
+  noStroke();
+  frameRate(40);
+  colorMode(HSB, 255);
+  intcond();
+}
+
+int t=0;
+
+void intcond()
+{
+\\ initial conditions of the system
+  fill(255);
+  
+ float h;
+  h= height/bh;	
+  float w;
+  w= width/bw;
+  
+ for(int i=0; i < n; i+=1)
+  {
+    pos_x[i]=int(random(width/2))+width/4;   
+    pos_y[i]=int(random(height/2))+height/4;
+    vel_x[i]=0.01*(int(random(4))-2)+200*(width/2-pos_y[i]+1)/(pos_x[i]*pos_x[i]+pos_y[i]*pos_y[i]); 
+    vel_y[i]=0.01*(int(random(4))-2)-200*(height/2-pos_x[i]+1)/(pos_x[i]*pos_x[i]+pos_y[i]*pos_y[i]);   
+    mass[i]=int(random(4));
+  }
+}
+
+void draw() 
+{ 
+  fill(255);
+  rect(0,0,width,height);
+  
+  int[][] adjust;
+ float h;
+  h= height/bh;
+  float w;
+  w= width/bw;
+  
+  for(int i=0; i < n; i+=1)
+  {
+    float dv_x = 0;
+    float dv_y = 0;
+    for(int j=0; j < n; j+=1){
+    \\equation of motion
+    	 dv_x = dv_x - a*(pos_x[i] - pos_x[j])*mass[i]*mass[j];
+    	 dv_y = dv_y - a*(pos_y[i] - pos_y[j])*mass[j]*mass[i];
+   
+	 pos_x[i] = pos_x[i] + vel_x[i]/20;
+    	 pos_y[i] = pos_y[i] + vel_y[i]/20;
+    	 vel_x[i] = vel_x[i] + dv_x;
+    	 vel_y[i] = vel_y[i] + dv_y;
+    	 if(pos_x[i]-pos_x[j] < mass[i]/2 & pos_x[i]-pos_x[j]>0){
+   	       if(pos_y[i]-pos_y[j] < mass[i]/2 & pos_y[i]-pos_y[j]>0){
+      	           mass[i]=mass[i]+mass[j];
+      		   mass[j]=0;
+      		   vel_x[i]= (mass[i]*vel_x[i]+mass[j]*vel_x[j])/(mass[i]+mass[j]);
+      		   vel_y[i]= (mass[i]*vel_y[i]+mass[j]*vel_y[j])/(mass[i]+mass[j]);
+   		    }
+   	 }
+    }
+  }
+  float x_a = 0;
+  float y_a = 0;
+  int U = 0;
+  for(int i=0; i<n; i+=1){
+    if(mass[i]>0){
+      U+=1;
+      x_a = x_a + pos_x[i];
+      y_a = y_a + pos_y[i];
+    }
+  }
+  float x_av = x_a/U;
+  float y_av = y_a/U;
+  \\draw the boxes
+  for(int i=0; i<n; i+=1){
+  if(pos_x[i]<width){
+      if(pos_y[i]<height){
+        if(mass[i]>0){
+        int x = int(pos_x[i] - (width/2 - x_av));
+        int y = int(pos_y[i] - (height/2 - y_av));
+        fill(5*i, 255, 255);
+        rect(x,y,2*mass[i],2*mass[i]);
+        }
+      }
+    }
+  }
+}
+
+

poincarebendixson.pde

@@ -0,0 +1,63 @@
+//Demonstration of the Poincare Bendixson Theorem
+//Eventually might be expanded to identify such limit cycles in a given set of equations.
+
+// Ben Horowitz ([email protected])
+
+
+//block resolution
+int bh = 34;
+int bw = 34;
+
+void setup()
+{
+  size(600,600);
+  fill(255);
+  //noStroke();
+  colorMode(HSB, 255);
+}
+
+float x;
+float[][] xp= new float[bw][bh];
+float y;
+float[][] yp= new float[bw][bh];
+
+void arrow(int x1, int y1, int x2, int y2) {
+  line(x1, y1, x2, y2);
+  float d = min(dist(x1, y1, x2, y2), 40);
+  pushMatrix();
+  translate(x2, y2);
+  float a = atan2(x1-x2, y2-y1);
+  rotate(a);
+  line(0, 0, -d/4, -d/4);
+  line(0, 0, d/4, -d/4);
+  popMatrix();
+}
+
+void draw()
+{
+  stroke(0);
+  background(120);
+  for(int i=0; i < bw; i+=1)
+  {
+    for(int j=0; j < bh; j+=1)
+    {
+      x=(i-bw*0.5)/8;
+      y=(j-bh*0.5)/8;
+      float xp =(-1*y+x*(1-x*x-y*y));
+      float yp = (1*x+y*(1-x*x-y*y));
+      int xt=i*width/bw;
+      int yt=j*height/bh;
+      //float nor=(sqrt((xtp)*(xtp)+(ytp)*(ytp)));
+      
+      int l;
+      l =int( width/(sqrt(bw*bw+bh*bh)))+2;
+
+      arrow(xt,height-yt,xt+int(10*xp),height-(yt+int(10*yp)));
+    }
+  }
+   noFill();
+   stroke(180);
+      ellipse(width/2,height/2, 4*bw,4*bh);
+      ellipse(width/2,height/2, 16*bw,16*bh);
+}
+

wave.pde

@@ -0,0 +1,109 @@
+// Wave equation with a fixed boundary condition
+// Ben Horowitz ([email protected])
+
+int bh = 100;
+int bw = 100;
+float[][] temp = new float[bh][bw];
+float[][] tm1 = new float[bh][bw];
+float[][] tm2 = new float[bh][bw];
+float k = 0.5;
+
+
+ 
+void setup()
+{
+  size(600,600);
+  fill(255);
+  noStroke();
+  //frameRate(1);
+  colorMode(HSB, 255);
+  intcond();
+}
+
+int t;
+
+boolean incircle(float i, float j)
+{
+  
+   float h;
+  h= height/bh;
+  float w;
+  w= width/bw;
+  
+  float d= min(height,width);
+ 
+ return( ((width/2-j*w)*(width/2-j*w) + (height/2-i*h)*(height/2-i*h))<((d/2-10)*(d/2-10)));
+  
+}
+  
+
+void intcond()
+{
+ 
+ for(int i=0; i < bw; i+=1)
+  {
+      for(int j=0; j < bh; j+=1)
+      {
+       
+      if(!incircle(i,j))
+      {
+        tm2[i][j]=tm1[i][j]=128;
+      }
+      /*
+      else if(i!=0 && i!=bw-1 && j==0 ||  j==bh-1)
+      {
+        tm2[i][j]=tm1[i][j]=128;
+      }
+      */
+      else
+      {
+   
+      tm2[i][j]=tm1[i][j]=10;
+      }
+      }
+  }
+}
+
+void draw() 
+{ 
+  int[][] adjust;
+ float h;
+  h= height/bh;
+  float w;
+  w= width/bw;
+  t=t+1;
+  
+  for(int i=0; i < bw; i+=1)
+  {
+    for(int j=0; j < bh; j+=1)
+    {
+      float x;
+      float a;
+     
+      
+      if(!incircle(i,j))
+      {
+        temp[i][j] = tm1[i][j];
+      }
+      else
+      {
+      //a= (tm1[i+1][j]+tm1[i-1][j]+tm1[i][j+1]+tm1[i][j-1])/4;
+      temp[i][j] = 2*tm1[i][j] - tm2[i][j] +  (tm1[i+1][j]+tm2[i-1][j]+tm1[i][j+1]+tm2[i][j-1]-4*tm1[i][j]) /4;
+      }
+      float value = temp[i][j];  
+      fill(100, 0, value);
+      //fill(value, 255, 255);
+      rect(i*w,j*h,w,h);
+      tm2[i][j]=tm1[i][j];
+      tm1[i][j]=temp[i][j];
+    }
+    /*
+    float[][] swap = tm2;
+    tm2 = tm1;
+    tm1 = temp;
+    temp = swap;
+    */
+  }
+
+} 
+