Dynamics of interacting particles
From Department of Theoretical and Applied Mechanics
Virtual laboratory > Dynamics of interacting particles
Here is the latest version of the program, which simulates the dynamics of interacting particles. Each particle is a viscoelastic sphere. The interaction between the spheres is described by the Lennard-Jones potential
Гравитация:
mg =
⋅ m ⋅ g0
Сколько шаров помещается по вертикали:
Конфигурация:
Short Lennard-Jones potential
Термостат:
T ⋅ T0 =
Разгон случайными скоростями
Термостат действует на: Внешнее трение Внутреннее трение
Разгон случайными скоростями
Термостат действует на: Внешнее трение Внутреннее трение
T ≈
Количество частиц:
Download program: Balls_v6_release.zip
The text of the program is written in JavaScript (developed by Anton Krivtsov, Tsvetkov Denis):
File "Balls_v6_release.js"
1 function MainBalls(canvas, slider_01, text_01, slider_02, text_02) {
2
3 canvas.onselectstart = function () {return false;}; // lock allocation canvas
4
5 // Starting parameters
6
7 var context = canvas.getContext("2d"); // context for drawing
8 canvas.oncontextmenu = function (e) {return false;}; // lock context menu
9
10 var Pi = 3.1415926; // "pi"
11
12 var m0 = 1; // weight
13 var t0 = 1; // Time (period of fluctuation)
14 var a0 = 1; // length (sphere's diameter)
15
16 var g0 = a0 / t0 / t0; // acceleration
17 var k0 = 2 * Pi / t0; // frequency
18 var C0 = m0 * k0 * k0; // rigidity
19 var B0 = 2 * m0 * k0; // viscosity
20
21 // *** Physical parameters ***
22
23 var Ny = 5; // The number of the spheres which are placed vertically
24 var m = 1 * m0; // weight
25 var CWall = 10 * C0; // wall's rigidity
26 var CBall = 0.1 * CWall; // rigidity between spheres
27 var BVisc = 0.008 * B0; // viscosity of the environment
28 var BInternal = 0.01 * B0; // internal friction
29 var BWall = 0.03 * B0; // viscosity of the walls
30 var mg = 0.25 * m * g0; // gravitational force
31 var r = 0.5 * a0; // radius of the sphere
32 var K = 0.7; // forces depending on radius
33 var a = 2 * r; // equilibrium state between particles
34 var aCut = 2 * a; // cutting radius
35 var TGoalK = 2; // temperature of the system TGoalK * D
36 var TActualMaxK = 200; // maximum temperature when thermostat work
37
38 // *** Calculation parameters ***
39
40 var fps = 60; // frames per second
41 var spf = 100; // steps per frame
42 var dt = 0.04 * t0 / fps; // integration step
43
44 // Running program
45
46 var r2 = r * r; // optimizing
47 var a2 = a * a; // optimizing
48 var D = a2 * CBall / 72; // bond's energy between spheres
49 var LJCoeff = 12 * D / a2; // coefficient for Lennard-Jones potential
50 var b = Math.pow(13 / 7, 6) * a; // coefficient for SLJ potential
51 var b2 = b * b; // optimizing
52 var SLJDenominator = 1 / (aCut * aCut - b2); // denominator for calculation SLJ potential
53
54 var thermostatEnabled = document.getElementById('checkbox_02').checked; // thermostat for viscosity of the environment
55 var addRandomV = document.getElementById('checkbox_03').checked; // random speed for acceleration of the spheres
56 var T0 = 1 * D; // temperature
57 var TGoal = TGoalK * T0; // goal temperature
58 var TActualMax = TActualMaxK * T0; // maximum temperature for thermostat
59 var TActual = 0; // actual temperature
60 var k = 1; // Boltzmann constant
61 var Tk = m / k; // optimizing
62 var viscFrictionTh = document.getElementById('checkbox_04').checked; // application thermostat for the viscosity environment
63 var internalFrictionTh = document.getElementById('checkbox_05').checked; // application thermostat for the internal friction
64
65 var Ka = K * a; // optimizing
66 var K2a2 = K * K * a2; // optimizing
67
68 var dNd = null; // taken sphere by cursor (drag & drop)
69 var grad; // gradient
70 var SLJEnabled = document.getElementById('checkbox_01').checked;
71
72 this.setSlider_01 = function(c) {mg = c * m * g0;}; // gravitation function
73 this.setSlider_02 = function(c) {TGoal = c;}; // thermostat function
74 this.setNy = function(ny) {
75 Ny = ny;
76 if (Ny > 8) {
77 grad = false; // gradient doesn't work if Ny > 8
78 context.fillStyle = "#3070d0"; // sphere's colour
79 } else
80 grad = true;
81 };
82 this.setNy(Ny); // start
83 this.setCheckbox_01 = function(bool) {SLJEnabled = bool;};
84 this.setCheckbox_02 = function(bool) {
85 thermostatEnabled = bool;
86 document.getElementById('checkbox_03').disabled = !bool;
87 document.getElementById('checkbox_04').disabled = !bool;
88 document.getElementById('checkbox_05').disabled = !bool;
89 document.getElementById('slider_02').disabled = !bool;
90 document.getElementById('text_02').disabled = !bool;
91 if (bool) {
92 TempIntervalID = setInterval( // update informtion about temperature
93 function(){document.getElementById('Temperature').innerHTML = TActual.toFixed(3);}, 1000 / 3);
94 }
95 else {
96 clearInterval(TempIntervalID); // delete updating informtion about temperature
97 document.getElementById('Temperature').innerHTML = "???"
98 }
99 };
100 this.setCheckbox_02(thermostatEnabled); // start for updating parameters
101 this.setCheckbox_03 = function(bool) {addRandomV = bool;};
102 this.setCheckbox_04 = function(bool) {viscFrictionTh = bool;};
103 this.setCheckbox_05 = function(bool) {internalFrictionTh = bool;};
104
105 // Setup of the interface
106
107 slider_01.min = 0; slider_01.max = 5;
108 slider_01.step = 0.05;
109 slider_01.value = mg / m / g0; // the first position of the slider
110 text_01.value = mg / m / g0;
111 slider_02.min = 0; slider_02.max = 5;
112 slider_02.step = 0.05;
113 slider_02.value = TGoal; // the first position of the slider
114 text_02.value = TGoal.toFixed(1);
115
116 // Start of the new system
117
118 // the following variables have to be recalculated every time when we change value Ny
119 var scale, w, h;
120 var rScale13, rScaleShift;
121 this.newSystem = function() {
122 scale = canvas.height / Ny / a0; // coefficient for transition from settlement to screen coordinates
123 w = canvas.width / scale; // window width in settlement coordinates
124 h = canvas.height / scale; // window height in settlement coordinates
125
126 rScale13 = r * scale * 1.3; // optimizing
127 rScaleShift = r * scale / 5; // optimizing
128
129 this.setRandom(); // set random configuration
130 };
131
132 // Работа с мышью
133
134 var mx_, my_; // cursor position
135
136 canvas.onmousedown = function(e) { // function for press on the cursor
137 var m = mouseCoords(e); // coordinates for calculation of the cursor position
138 //return cycle
139 for (var i = balls.length - 1; i >= 0; i--) {
140 var b = balls[i];
141 var rx = b.x - m.x;
142 var ry = b.y - m.y;
143 var rLen2 = rx * rx + ry * ry; // length between cursor and sphere
144 if (rLen2 <= r2) { // the cursor has pressed on a sphere
145 if (e.which == 1) { // mouse's left button has pressed
146 dNd = b;
147 dNd.xPlus = dNd.x - m.x; // shift of the cursor on X
148 dNd.yPlus = dNd.y - m.y; // shift of the cursor on Y
149 mx_ = m.x; my_ = m.y;
150 canvas.onmousemove = mouseMove; // shift function work while button press
151 } else if (e.which == 3) // mouse's right button has pressed
152 balls.splice(i, 1); // delete sphere
153 return;
154 }
155 }
156
157 // if don't exit - add new sphere
158 if (e.which == 1) {
159 dNd = addNewBall(m.x, m.y, true); // add sphere and take it
160 if (dNd == null) return; // return if the ball doesn't add
161 dNd.xPlus = 0; dNd.yPlus = 0; // set ball in center position
162 mx_ = m.x; my_ = m.y;
163 canvas.onmousemove = mouseMove; // shift function works while mouse's button has pressed
164 }
165 };
166
167 document.onmouseup = function(e) { // function when mouse's button let off
168 canvas.onmousemove = null; // shift function equal null when mouse's button let of
169 dNd = null; // ball doesn't exist when mouse's button let of
170 };
171
172 function mouseMove(e) { // shift function when left mouse's button pressed
173 var m = mouseCoords(e); // coordinates for calculation
174 dNd.x = m.x + dNd.xPlus;
175 dNd.y = m.y + dNd.yPlus;
176 dNd.vx = 0.6 * (m.x - mx_) / dt / fps; dNd.vy = 0.6 * (m.y - my_) / dt / fps;
177 mx_ = m.x; my_ = m.y;
178 }
179
180 function mouseCoords(e) { // function returns settlement coordinates of the cursor of a mouse
181 var m = [];
182 var rect = canvas.getBoundingClientRect();
183 m.x = (e.clientX - rect.left) / scale;
184 m.y = (e.clientY - rect.top) / scale;
185 return m;
186 }
187
188 // Работа с массивом
189
190 var balls = []; // sphere's massive
191 var addNewBall = function(x, y, check) {
192 // check - whether is crossed a new sphere with walls or already existing spheres
193 if (check) {
194 if (x - r < 0 || x + r > w || y - r < 0 || y + r > h) return null;
195 for (var i = 0; i < balls.length; i++) {
196 var rx = balls[i].x - x;
197 var ry = balls[i].y - y;
198 var rLen2 = rx * rx + ry * ry;
199 if (rLen2 < 4 * r2) return null;
200 }
201 }
202
203 var b = [];
204
205 b.x = x; b.y = y; // calculation cordinates
206 b.fx = 0; b.fy = mg; // force
207 b.vx = 0; b.vy = 0; // velocity
208
209 balls[balls.length] = b; // add an element in end of massive
210 return b;
211 };
212
213 this.setEmpty = function() {balls = [];}; // free area
214
215 this.setRandom = function() { // random configuration
216 balls = [];
217 for (var i = 0; i < 1000; i++)
218 addNewBall(Math.random() * w, Math.random() * h, true);
219 };
220
221 var sqrt3 = Math.sqrt(3);
222 this.setTriangularLattice = function() { // triangle lattice
223 balls = [];
224 var center = (w - Math.floor(w / r) * r) / 2; // shift
225 for (var j = 0; j < Math.floor(h / (sqrt3 * r)); j++)
226 for (var i = 0; i < Math.floor(w / r) - 1; i++)
227 if ((i + j) % 2 == 0) addNewBall(r * (i + 1) + center, h - r * (1 + sqrt3 * j), false);
228 };
229
230 // program's cycle
231
232 function control() {
233 physics();
234 draw();
235 }
236
237 // calculation part
238
239 function physics() {
240 for (var s = 1; s <= spf; s++) {
241
242 var BViscTh = BVisc;
243 var BInternalTh = BInternal;
244 // работа термостата
245 if (thermostatEnabled) {
246 if (balls.length > 0) {
247 var v2Sum = 0;
248 for (var i1 = 0; i1 < balls.length; i1++)
249 v2Sum += balls[i1].vx * balls[i1].vx + balls[i1].vy * balls[i1].vy;
250 var v2Average = v2Sum / balls.length;
251 TActual = Tk * v2Average;
252
253 if (addRandomV) { // random velocities if the temperature is low
254 if (TGoal > 0.15 && TActual < 0.1) {
255 for (var i2 = 0; i2 < balls.length; i2++) {
256 balls[i2].vx += 0.3 * (1 - 2 * Math.random());
257 balls[i2].vy += 0.3 * (1 - 2 * Math.random());
258 }
259 }
260 }
261
262 if (TActual < TActualMax) { //because of the fact that a mouse spheres can set ultraboundary speed
263 if (viscFrictionTh) BViscTh = BVisc * (TActual - TGoal); // thermostat
264 if (internalFrictionTh) BInternalTh = BInternal * (TActual - TGoal); // thermostat
265 }
266 } else
267 TActual = 0; // for thermostat on page
268 }
269
270 //recalculation of forces goes the certain massif since forces of interaction between spheres will be added further
271 for (var i0 = 0; i0 < balls.length; i0++) {
272 balls[i0].fx = - BViscTh * balls[i0].vx;
273 balls[i0].fy = mg - BViscTh * balls[i0].vy;
274 }
275
276 for (var i = 0; i < balls.length; i++) {
277 // calculation of interaction is made with all following spheres in the massif not to consider each interaction twice
278 var b = balls[i];
279 for (var j = i + 1; j < balls.length; j++) {
280 var b2 = balls[j];
281 var rx = b.x - b2.x; var ry = b.y - b2.y; // vector on the first ball (b)
282 var r2 = rx * rx + ry * ry; // length between spheres
283 var rLen = (Math.sqrt(r2));
284 if (rLen > aCut) continue; // check for cutting radius
285
286 // if the distance between particles isn't enough, forces will be counted like K * a
287 if (rLen < Ka) {
288 if (rLen > 0.00001) { // check
289 rx = rx / rLen * Ka;
290 ry = ry / rLen * Ka;
291 }
292 r2 = K2a2;
293 rLen = Ka;
294 }
295
296 // сила взаимодействия
297 var s2 = a2 / r2; var s4 = s2 * s2; // optimizing
298 var F = LJCoeff * s4 * s4 * (s4 * s2 - 1); // LJ interaction force
299 if (SLJEnabled) {
300 var kSLJ; // k(r) - coefficient SLJ potential
301 if (r <= b) kSLJ = 1;
302 else {
303 var brackets = (r2 - b2) * SLJDenominator;
304 kSLJ = 1 - brackets * brackets;
305 }
306 F *= kSLJ;
307 }
308
309 // internal friction between spheres
310 if (r2 < a2) {
311 var vx21 = b.vx - b2.vx; var vy21 = b.vy - b2.vy; // vector on the first ball (b)
312 var ex = rx / rLen; var ey = ry / rLen;
313 var v = vx21 * ex + vy21 * ey;
314 F -= F * BInternalTh / rLen * v;
315 }
316
317 // summirize forces
318 var Fx = F * rx; var Fy = F * ry;
319
320 b.fx += Fx; b.fy += Fy;
321 b2.fx -= Fx; b2.fy -= Fy;
322 }
323
324 if (b == dNd) continue; // if the sphere is grabbed with the cursor - his interaction with walls and we don't consider movement
325
326 if (b.y + r > h) { b.fy += -CWall * (b.y + r - h) - BWall * b.vy; }
327 if (b.y - r < 0) { b.fy += -CWall * (b.y - r) - BWall * b.vy;}
328 if (b.x + r > w) { b.fx += -CWall * (b.x + r - w) - BWall * b.vx; }
329 if (b.x - r < 0) { b.fx += -CWall * (b.x - r) - BWall * b.vx; }
330
331 b.vx += b.fx / m * dt; b.vy += b.fy / m * dt;
332 b.x += b.vx * dt; b.y += b.vy * dt;
333 }
334 }
335 }
336
337 // drawing
338 function draw() {
339 context.clearRect(0, 0, w * scale, h * scale); // clear
340 for (var i = 0; i < balls.length; i++){
341 var xS = balls[i].x * scale; var yS = balls[i].y * scale;
342 if (grad) {
343 // calculation of a gradient needs to be carried out for each sphere
344 var gradient = context.createRadialGradient(xS, yS, rScale13, xS - rScaleShift, yS + rScaleShift, 0);
345 gradient.addColorStop(0, "#0000bb");
346 gradient.addColorStop(1, "#44ddff");
347 context.fillStyle = gradient;
348 }
349
350 context.beginPath();
351 context.arc(xS, yS, r * scale, 0, 2 * Math.PI, false);
352 context.closePath();
353 context.fill();
354 }
355 }
356
357 // Start
358 this.newSystem();
359 setInterval(control, 1000 / fps);
360 // function updates information on quantity of particles in the field
361 setInterval(function(){document.getElementById('ballsNum').innerHTML = balls.length;}, 1000 / 20);
362 }
Файл "Balls_v6_release.html"
1 <!DOCTYPE html>
2 <html>
3 <head>
4 <meta charset="UTF-8" />
5 <title>Balls</title>
6 <script src="Balls_v6_release.js"></script>
7 </head>
8 <body>
9 <canvas id="canvasBalls" width="800" height="600" style="border:1px solid #000000;"></canvas>
10 <br>
11 <div>Gravitation:
12 <input type="range" id="slider_01" style="width: 150px;" oninput="app.setSlider_01(this.value); document.getElementById('text_01').value = this.value;">
13 mg =
14 <input id="text_01" style="width: 5ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
15 // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
16 if (!this.checkValidity()) return;
17 app.setSlider_01(this.value);
18 document.getElementById('slider_01').value = this.value;
19 ">
20 ⋅ m ⋅ g0</div><br>
21
22 <div>The number of the spheres which are placed vertically:
23 <input type="button" style="width: 30px" name="" onclick="app.setNy(3); app.newSystem();return false;" value="3"/>
24 <input type="button" style="width: 30px" name="" onclick="app.setNy(4); app.newSystem();return false;" value="4"/>
25 <input type="button" style="width: 30px" name="" onclick="app.setNy(5); app.newSystem();return false;" value="5"/>
26 <input type="button" style="width: 30px" name="" onclick="app.setNy(7); app.newSystem();return false;" value="7"/>
27 <!--знак пробела-->
28 <input type="button" style="width: 30px" name="" onclick="app.setNy(9); app.newSystem();return false;" value="9"/>
29 <input type="button" style="width: 30px" name="" onclick="app.setNy(12); app.newSystem();return false;" value="12"/>
30 <input type="button" style="width: 30px" name="" onclick="app.setNy(15); app.newSystem();return false;" value="15"/>
31 </div><br>
32
33 <div>Configuration:
34 <input type="button" name="" onclick="app.setTriangularLattice(); return false;" value="Triangular lattice"/>
35 <input type="button" name="" onclick="app.setRandom(); return false;" value="Systemless"/>
36 <input type="button" name="" onclick="app.setEmpty(); return false;" value="A blank field "/>
37 </div><br>
38
39 <div>
40 <input type="checkbox" id="checkbox_01" name="" onchange="app.setCheckbox_01(this.checked);"/>
41 <a href="/SLJ" title="SLJ" class="mw-redirect">Short Lennard-Jones</a> potential
42 </div><br>
43
44 <div>
45 <input type="checkbox" id="checkbox_02" name="" onchange="app.setCheckbox_02(this.checked);"/>
46 Thermostat:
47 <input type="range" id="slider_02" style="width: 150px;" oninput="app.setSlider_02(this.value); document.getElementById('text_02').value = this.value;">
48 T ⋅ T0 = <input id="text_02" style="width: 5ex;" required pattern="[-+]?([0-9]*\.[0-9]+|[0-9]+)" oninput="
49 // if not the number is entered - the line won't pass validation on a pattern above, and checkValidity () will return false
50 if (!this.checkValidity()) return;
51 app.setSlider_02(this.value);
52 document.getElementById('slider_02').value = this.value;
53 ">
54 <br>
55 <input type="checkbox" checked id="checkbox_03" name="" onchange="app.setCheckbox_03(this.checked);"/>Acceleration is achieved by random rates
56 <br>
57 The thermostat acts on:
58 <input type="checkbox" checked id="checkbox_04" name="" onchange="app.setCheckbox_04(this.checked);"/>External friction
59 <input type="checkbox" checked id="checkbox_05" name="" onchange="app.setCheckbox_05(this.checked);"/>Internal friction
60 <div>T ≈ <span id="Temperature"></span></div>
61 </div><br>
62
63 <div>Number of the particles: <span id="ballsNum"></span></div>
64
65 <script type="text/javascript">var app = new MainBalls(
66 document.getElementById('canvasBalls'),
67 document.getElementById('slider_01'),
68 document.getElementById('text_01'),
69 document.getElementById('slider_02'),
70 document.getElementById('text_02')
71 );</script>
72 </body>
73 </html>
Here you can find the previous versions of the program.
The proposed development directions
- Add temperature regulation by Hoover‘s thermostat.
- The program's evaluation speed in order to optimize it.
- Apply the copying principle.
- Add different potentials.
- Add the ability to set viscoelastic walls anywhere.
- Add the periodic boundary conditions.
