Revision History Date Version Description Author 03/06/ Initial Version Sergio Miguel Martin. Table of Contents

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1 Icarus Physics Engine Manual 1 - Basic Engine Operation Version 1.0 Revision History Date Version Description Author 03/06/ Initial Version Sergio Miguel Martin Table of Contents 1. Introduction Purpose Scope Definitions, Acronyms and Abbreviations References 2 2. Introduction to the finite differential method of solving complex problems Introduction to complex problems Integral methods limitations Finite differential Method How does the Icarus Physics Engine work? Basic IPE operation Impact of accumulated error on the simulation Gravitation, Electric and Magnetic forces Gravitation Forces Electric Forces Magnetic Forces Time differential commit operation Optimization methods Non Optimized Mode n-step Euler Method Runge-Kutta Method 23 Universidad Nacional de La Matanza Page 1 de 24

2 Manual 1 - Basic Engine Operation 1. Introduction 1.1 Purpose The purpose of this document is to introduce the reader to the complexity of the continuous problem resolution, the differential approach solution, define the basic differential algorithms and procedures used in the Icarus Physics Engine, and the numeric optimization methods implemented. 1.2 Scope This document describes first the mathematical approach to the field simulations problems and then show the overall operation and algorithms to be used by Icarus Physics Engine in order to simulate a object / fields oriented schema under Gaussian fields (limited to: gravitation and electromagnetic fields) 1.3 Definitions, Acronyms and Abbreviations IPE Icarus Physics Engine 1.4 References N/A Universidad Nacional de La Matanza Page 2 de 24

3 2. Introduction to the finite differential method of solving complex problems 2.1 Introduction to complex problems The Icarus Physics Engine will simulate gravitation scenarios where n-quantity of objects will interact in real time. As their position varies so will their gravitation field and its influence in other objects. So, is there a magic formula to determine the whole system state in a certain moment of the future? Well, modern physics and math count with a wide practical knowledge on solving any kind systems. But they will only solve simplified versions of those systems as they provide a tool for an specific area of science. Complex systems (such as the ones to simulate in Icarus) will require that it can be divided into a discrete quantity of simpler systems able to be solved with these tools. As example, when a student is taught about gravitational forces, it s exemplified by simple exercises that discard factors other than a constant gravity acceleration itself by considering them despicable (and they are: air resistance, gravity variation thorough altitude). This way the real complex system becomes a singular simple system able to be calculated using simple formulas. This methodology is very useful to teach and learn this phenomenon, but often fall short in front of more complex systems that (unfortunately) can not be solved with the application of a handful of formulas. Figure 1: Simple free fall example. In the Diagram 1 a standard free fall exercise example is shown. While a gravitation force (shown in black) pulls down the object to the earth s surface, other factors such as Air Resistance and the much less obvious gravitation field variation effects (shown in red) on the studied case are considered despicable. Now, while the fact that its influence in the final development of the exercise is practically despicable is the only reason to discard Air Resistance, it is the partial truth when gravitation field variation comes to play its cards. The second factor for gravitation field variation to not being taken account is that its continuous solution has a very complex solution in some cases and, in some other, no mathematical solution at all. Universidad Nacional de La Matanza Page 3 de 24

4 The gravitation acceleration over any existing thing over the universe varies in the inverse of the square of the distance with the attracting mass. So, reviewing Diagram 1, we can point that at Altitude 1, earth s gravitation acceleration is greater from Altitude 2 as much as the inverse of the square of the increment (distance) allows. This distance difference (at the scale of meters) has such small effect in acceleration that it would affect very little in the final duration of the fall. So little that it can be ignored. g 9.8m / s Formula 1: Gravity remains constant thorough the whole exercise 2 But is it so for larger scales of distance? The answer is definitely no. The linear gravity scalar accelerations for different distances (in crescent order) from the earth are shown in the following table: Where Distance from Earth s core Gravity Acceleration At the surface of the earth (poles): 6356 km 9,8322 m/s 2 At the surface of the earth (equator): 6378 km 9,78 m/s 2 MIR Space station 6656 km 8.18 m/s 2 Termosphere 7356 km 7.32 m/s 2 Geostationary orbit km 0,3113 m/s 2 Moon s average orbit km 0,0027 m/s 2 Table 1: Gravitation acceleration in different distances from earth As shown Table 1. The earth s gravitation influence in nearby bodies greatly varies at kilometrical magnitudes. Moreover, at the distance of ten thousand kilometers you experience half of the gravity acceleration that you would if you were at the very surface. So, now imagine a free fall example from the Moon s average orbit distance. You would take a lot longer to finally touch earth s surface (we strongly recommend using a parachute, though) than what you could calculate using the simplified theoretical exercises where the gravity is a constant equal to 9.8m/s. You would start feeling a slow acceleration (0,0027 m/s 2 ) which will fast increase as you pass close to the MIR Space station (8.18 m/s 2 ) and then get close to the surface (9,8 m/s 2 ). At the end of this experience, the elapsed time would be much more than expected as acceleration started much slower than the one at surface. Universidad Nacional de La Matanza Page 4 de 24

5 Figure 2: Formula complexity for gravitation exercises using variable gravity force. Now, there is nothing to criticize to our physics professors. Formulas for free fall (see Figure 2) calculations keeping in mind the gravity acceleration variation are extremely complex. If now we add other factors of complexity such as another dimension (moving sideways while falling as on Case 2) or add multiple sources of gravity (imagine an object at the middle of moon and earth being attracted by both as in Case 3), there is no single step solution that could possibly calculate this situation. Fortunately, there is a way to get to a case 3-like (or even more complex) solution, but first we will analyze why the current approach is unable to do this. Universidad Nacional de La Matanza Page 5 de 24

6 2.2 Integral methods limitations When we first try to solve a free-fall problem such as the one in figure 1 to calculate how long would take for an object falling from, as example, 50 meters high to reach the earth s ground, we use the following well-known formulas: Where: Formula 2: Standard free-fall calculation formulas is the initial velocity (m/s). is the vertical velocity with respect to time (m/s). is the initial altitude (m). is the altitude with respect to time (m). is time elapsed (s). is the constant acceleration due to gravity (set to 9.81 m/s 2 ). Given that the acceleration is taken as constant thorough the whole exercise duration, time difference has no effect on this value, therefore making it an independent variable respecting to time. Figure 3: Object s acceleration thorough the exercise From the first formula, we can see that final velocity given a certain variation of time, depends linearly on the gravity s acceleration. In other words, we can define the velocity as a variable that is defined integration of the gravity s acceleration thorough the variation of time. Universidad Nacional de La Matanza Page 6 de 24

7 In the figure 3 we can define the unsigned amount of the velocity variation as the area enclosed by the gravity constant and the start/end times. Geometrically we can define this as the rectangle s area where the height is the gravity constant and the length is the time elapsed. This we can verify that the variation of the velocity obeys the formula 2: ( 1) v gt g * t v0 (2) v g * t Formula 3: Absolute variation of velocity depends linearly on gravity This shows that, as time flows from start to end, velocity will have to vary linearly, slowing its way up, completely stopping at the highest point, and accelerating to its way to the floor. We can see this graphically in the Figure 4: Figure 4: Object s Velocity thorough the exercise In the figure 4 we can define the unsigned amount of the altitude variation as the area enclosed by the linear velocity variation and the start/end times. This means that the variation of altitude can be defined as the integral of the velocity respecting to time ( 3) h vt v * t h Formula 3: Absolute variation of altitude depends linearly on velocity 0 But since velocity is not constant, but linearly dependent of the gravity s acceleration, we can compose the amount of the velocity s variation from the (2) equation into the v value of (3) as the double integral of the gravity respecting to time twice: Universidad Nacional de La Matanza Page 7 de 24

8 (4) h gt t gt v 0 t gt (5) h 2 2 v t h Formula 4: Absolute variation of position depends on the square of the gravity 0 0 Now we can see the evolution of the position in the following graph in figure 5: Figure 5: Altitude finally depends on the square of the time At this point, having gravitation as constant, we have been able to solve a simple free fall exercise using integral methods. The unknown variable must be put alone at once side of the (5) equation and the calculation of the other side will reveal its exact value. But this method has its limitations. As the problems become more and more complex (Cases 1 to 3), mathematical tools become less capable of determining a discrete quantity of formulas that could solve the system. In case 1, the gravitation s acceleration of the object, is not constant, but depends on the inverse of the square of the distance as in the following formula (6): Where g is the gravity s acceleration, G is the universal gravitation constant, Me is the earth s mass and d is the distance between the object and the earth. Universidad Nacional de La Matanza Page 8 de 24

9 e ( 6) g G 2 (7) d x M d earth x object Assuming that the object s mass is so little that it won t affect earth, the x position of the earth can be taken as a constant, this way, distance (d) will depend on the x position of the object: (8) (9) d f ( x (10) g object g f ( d) ) f ( x object ) Now, the gravity is not constant anymore, but depends on the position of the object (formula 10)! Since the position of the object will be affected, at its time, on the gravity in each point, the solution is not easy to obtain anymore. The gravitation force suffered by the object at each moment can be calculated with the distance from the attracting objects, but there is no way to know this value a priori using integral methods. This way, the only alternative for those and more complex cases is simplifying them using a discrete approach: the finite differential method. Universidad Nacional de La Matanza Page 9 de 24

10 2.3 Finite differential Method In the context of a case 1 One-Dimensional example, let s examine what happens to gravity s acceleration as the object approaches the earth: Figure 6: One-Dimensional example Universidad Nacional de La Matanza Page 10 de 24

11 Now that the gravity varies and depends on the distance, we can determine the strength of the gravitation field in any point of the space based on the distance from the attracting mass. Figure 7: Earth gravitational field s equipotential lines In the Figure 7, we can see that, at a certain radius, the value of the earth gravitational field remains equal. In this example, all the objects in the R1 radius will suffer the same gravitational attraction. At the same time, any object in the R1 radius will suffer more gravitational attraction than any object in the R2 radius. At its time, any object located in the R2 radius will suffer more gravitational attraction than any object in the R3 radius. From that, we are able affirm that: - In any position of space, there is one and only one value for the gravitational field and, (1) - This value can be obtained using a finite number of calculations (2) Remembering the first example where the gravitation acceleration was taken as constant (figure 1). The reasoning was that the variation of distance (in the order of mere meters) was not enough to significantly modify the results. So, if we can use the same formula 1 to calculate the position change of an object many times with a very little time differential recalculating the gravity value at each new position and taking it as constant, we can be able to reduce case 1, 2, 3 and even more complex problems into a finite number of simpler problems with a little cost which is information loss. To comprehend what we mean with information loss and how it is influenced by the time differential, we will give a graphical approach of the integration basics. Universidad Nacional de La Matanza Page 11 de 24

12 Let s define an arbitrary unknown function f(x) that depends on some x variable that encloses a given area (At) between its y value, the y = 0 axis, and side limits between x = 0 and x = x. (Remember figures 3 and 4 to see why determining the area below the function is important for our simulation). Figure 8: Arbitrary function f(x). A first approach to estimate the area enclosed by the function, we can draw series of rectangles based in the y value (based on the (2) affirmation) of the function at the x point where we are placed as height, and using a constant x differential (Δx) as length, as shown in the following Figure: Figure 9: Rectangular approximation to the total area using finite x differentials Universidad Nacional de La Matanza Page 12 de 24

13 As you can observe in the figure 9 we can approximate to the total area below the function by using a finite number of rectangles. The n times sum of the area of the rectangles will give an estimate of the area that will differ in an indeterminate error that we call information loss (E(i)). (11) At n i0 f ( it)* t E( i) In the next figure we can observe that, if we reduce the time differential (Δx), the number of steps (n) will increase, and the information loss will decrease. Figure 10: Information loss decreases as time differentials become smaller As logic consequence, as the time differential (Δx) approaches zero, steps number become infinite, and information loss approaches zero as well. As we can see in the following formula, this results in the definition of defined integral of the function: ( 12) At xx' x0 f ( x) dx lim Where an hypothetical E information loss would tend to zero. n n i1 f ( x ) x But using an infinite number of steps brings us to a new dead-end: n-quantity of steps would require an infinite computational power to calculate. We are back where we started and we don t have an exact function to define g. However, in the very first free fall example, we assumed that, for a few meters, the gravitational acceleration difference was so low that the information loss would not affect the result of the exercise in a significant way. Keeping that in mind, we can solve a complex problem where we lack a defining function for g (but can easily obtain its value in any moment of time), by dividing the whole experience time in very little time differentials taking a constant g to cover a few meters where it s time differential causes information loss so small to not affect the results significantly and updating the gravitation value in each step. And that s exactly how the Icarus Physics Engine works. Check the next chapter for more information on this. n Universidad Nacional de La Matanza Page 13 de 24

14 3. How does the Icarus Physics Engine work? 3.1 Basic IPE operation The Icarus Physics Engine (IPE), just as any other computational physics engine, uses the finite differential method to simulate the whole system in discrete time increments. IPE lets the user set the time differential in order to take it to a balance between acceptable error margins and computer performance. The following figure defines how the user can use the time differential as a balance between error (information loss) and performance (faster simulation): Figure 11: User can define a Time Differential that best fit his necessities IPE re-evaluates all force interactions to an object at the beginning of every step and obtains a resulting force which is the vector sum of all the other forces. Then uses this force to calculate vector acceleration and takes it as a constant. Once it has finished calculating this constant for all the objects in the system, applies the simple (5) formula using the obtained constant instead of g. Figure 12 illustrates the resulting forces on a given state: Figure 12: All the resulting forces are calculated before committing the step As result of the committing the formula (5) in each object of the system using the obtained acceleration as constant, the system passes to the following state (with a time equal to the previous state plus one time differential) where the object s positions and velocities are very close to the real case. (remember how close we can get to the real function by using the differential approach by reviewing the Figure 10). Universidad Nacional de La Matanza Page 14 de 24

15 In the following figure, we can see how we can approach a real orbit using finite series of linear steps (time differentials have been exaggerated to allow viewer note the linear form of the steps advances; all the embedded IPE scenarios have time differentials that are not even perceptible): Figure 13: Movement comparison between real and simulated orbits 3.2 Impact of accumulated error on the simulation While error (or information loss) in one step only may be despicable, its effect remains in for the next step. Even if we can consider that, in one step, the effects of gravity variation are despicable, as the steps run its effects accumulate and, in some future moment will end up in an error that could significantly affect the simulation s result. Reviewing Case 1 from Figure 2, we have a simple example of an object being attracted by another mass (which is not affected by the first object s mass, in order to simplify the example). See Figure 14: Universidad Nacional de La Matanza Page 15 de 24

16 Figure 14: Error accumulates as steps are simulated for a case 1 example Figure 14 shows how error due to setting gravity as constant accumulates step by step. Imagine that the error tolerated for this exercise was that the distance between the simulation and a real measure were not to exceed the 500 kilometers limit. After the first step, the error was only 50 kms, and was still tolerable for this simulation. After the second step, the combination between the previous error accumulation and the actual step s error made it increment to 200kms, a still tolerable value. But in the third step, the error accumulation produced a position error of 1000kms and the simulation could not be considered accurate anymore. Obviously, in this visual example, error was exaggerated. Modern computational capacity allows relatively fast simulations at an extremely low error rate. In some cases it could take very large amount of steps until the error accumulation could be considered significant. However, it s important to notice that, given that this is still a finite differential method, all simulations end up in significant error in some n-quantity of steps, independent of the time differential selected. There are, nonetheless, several ways to strongly reduce error accumulation using mathematical methods such as euler and runge-kutta methods which are implemented in IPE and can be selected by the user. These methods will be explained later in this manual. Universidad Nacional de La Matanza Page 16 de 24

17 IPE provides a way to estimate the absolute value of the error, and therefore evaluating if the simulation has suffered significant accumulation. Since we are simulating a closed system, energy must be conserved thorough the whole simulation. You can obtain the total system s energy as the sum of all the individual potential and kinetic energies at the beginning of simulation (when we know that error E is equal to zero). Then at each step you can recalculate the sum of those energies. Figure 15: Shadowed area shows possible deviations of the energy due to error accumulation As you can see on Figure 15, a real measurement is expected to show a constant energy thorough the whole experience time, but since the error in the simulation tends to accumulate, the total energy measurement tends to diverge as well. Energy divergence is then, a good indicative of error accumulation: E( t) T 0 T t Where E(t) is the error accumulation at a t moment, To is the total energy at the beginning of simulation and Tt is the total energy at the t moment. Universidad Nacional de La Matanza Page 17 de 24

18 4. Gravitation, Electric and Magnetic forces. The Icarus Physics Engine (IPE) is able to simulate gravitation, electric and magnetic interactions between the simulated bodies. It can even simulate a system where the three kind of forces (in this software, we deliberately separate electromagnetic force into two, for technical reasons), at the same time if these bodies have mass and charge and the user wants both of them to affect the simulation. The interactions between objects, uniform electric/magnetic, and electrically induced magnetic fields can be set as a list of rules, as example: Object Is affected by Through the following force Object 1 Object 2 Gravity Object 2 Object 1 Gravity Object 1 Magnetic Field 1 Magnetism This way, the user has the total control of the interactions included in the simulation. The way IPE simulates each force is explained in the following paragraphs: 4.1 Gravitation Forces The Icarus Physics Engine calculates the forced produced by the attraction between to bodies with mass in the space (x,y,z) using the Newton's law of universal gravitation. where: F is the magnitude of the gravitational force between the two point masses, G is the gravitacional constant* m 1 is the mass of the first point mass, m 2 is the mass of the second point mass, and r is the distance between the two point masses. *The default value for G is: And can be edited by the user. As IPE uses vector values for position, velocity and acceleration, the attraction force produced for its calculations must be obtained as a vector force as well, using the following formula: Universidad Nacional de La Matanza Page 18 de 24

19 4.2 Electric Forces The Icarus Physics Engine calculates the force produced by the attraction/repulsion between to charged bodies in the space (x,y,z) using the Coulomb's law. where: F is the magnitude of the electric force of attraction/repulsión between the two point charges, Ke is the Coulomb constant q 1 is the charge of the first particle, q 2 is the charge of the second particle, r is the distance between the two point masses and, is the unit vector pointing from the particle with charge q to the E-field evaluation point, *The default value for Ke is: And can be edited by the user. As IPE uses vector values for position, velocity and acceleration, the attraction/repulsion force produced for its calculations must be obtained as a vector force as well, using the following formula: Where Eo is the electric constant* *The default value for Eo is: A s/(v m) = F/m Universidad Nacional de La Matanza Page 19 de 24

20 4.3 Magnetic Forces The Icarus Physics Engine calculates the force produced by the action of a magnetic field (B) on a charged body (q) in the space (x,y,z) using the Lorentz force equation. where: B is the vector intensity of the magnetic field v is the vector velocity of the affected particle q is the charge of the affected particle, As IPE uses vector values for position, velocity and acceleration, the attraction/repulsion force produced for its calculations must be obtained as a vector force as well, and the Lorentz force equation provides it Note that, in difference with the other interactions where the force had the direction of the unit vector uniting both objects position, the magnetic field produces a force that is perpendicular both to the velocity and direction of the field. Icarus is an excellent way to learn about this phenomenon as you can clearly see the perpendicularity of the resulting force. 4.4 Time differential commit operation Once all the rules from the IPE list of rules were calculated, the resulting force (Fr) over any k object is the vector sum of all the individual gravitational/electric/magnetic forces from the other objects/fields working over this k object: n F k F ik i0 Having the resulting force on the object, the only thing we need to do to obtain the vector acceleration is by dividing it by the object s mass: a i Once we have all the acceleration vectors for all the objects in the simulation, IPE applies the formula 5 (using a as g) for each of them and commits the time differential. Fi m i Universidad Nacional de La Matanza Page 20 de 24

21 5. Optimization methods The Icarus Physics Engine allows the user to choose between three modes of solving finite differential simulations. 5.1 Non Optimized Mode This is the faster mode and takes only one calculation unit from each step. If the user chooses the non-optimized mode, the simulation will use the rectangular approach as seen on figure 9. The area under the field s function will be estimated by using a sum of the area of a finite number of rectangles whose lengths will be the time differential and their heights are the value of the function at the initial point. This mode produces much more information loss than the optimized modes and is recommended for simulations where the performance is much more important than the error accumulation. Figure 16: Non Optimized Mode 5.2 n-step Euler Method This is the most explicit and simple method for numeric integration for differential equations. The idea behind it is that we can use 2 or more calculation units per step to increase its precision with less information loss than with increasing the steps per unit of time (in other words, decreasing the time differential). We can generalize the Figure 16 case as a 1-step euler s method approximation. Now, instead of going ahead with the next time differential, we can apply another calculation unit using a 2-step euler s method. This is done by re evaluating the constant variable (acceleration) at the middle of the estimation and re calculating it. The result would be the following figure: Universidad Nacional de La Matanza Page 21 de 24

22 Figure 17: n-step Euler approximation method The idea behind the 2-step euler method is that, the acceleration is re-evaluated in function of the position that the object would have at the half of the time differential. Then a correction in the slope of the velocity will occur at Δt/2. In the 3-step euler method, the re-evaluation occurs two times, one at the third and the other at the second third of the time differential. A good balance between euler steps per time differential and time differential duration could greatly improve precision at very low performance cost. Universidad Nacional de La Matanza Page 22 de 24

23 5.3 Runge-Kutta Method This is the most efficient performance/accuracy method for numeric integration for differential equations. It is based on four steps, each one using the results obtained from the previous. Defining an initial problem as: The approximated solution is defined as: Where: This way, the following value (yn+1) is determined by the current value (yn) plus the product of an interval (h) and an estimated slope. This slope is obtained as the average of the following weighted slopes: k1 is the slope at the beginning of the interval. k2 is the slope at the middle of the interval, using k1 to determine the value of y in the Tn + h/2 point using euler s method k3 is again the slope evaluated at the middle of the interval but using k2 to determine the value of y k4 is the slope at the end of the interval using k3. Making an average of the four and assigning more importance to the two in the middle (k2 and k3), we obtain the runge-kutta estimated slope: The following figure shows graphically how the runge-kutta approximates to a very accurate slope in four steps. Universidad Nacional de La Matanza Page 23 de 24

24 Figure 18: Runge-Kutta Method Universidad Nacional de La Matanza Page 24 de 24