Runge-Kutta Method for Systems - Keystone Mining Post

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Differential Equations

~ RungeKutta-Fehlberg

DISCUSSION

MULTIRUNGE

METHOD
Higher-Order Runge-Kutta

In dealing with a system of coupled first-order differential equations, it is necessary to apply vector type operations (see menu under) "Vectors and Matrices". Although "Vectors and Matrices" is only a generalization of vector usage, the exact vector manipulation for this particular problem is not shown.

The concept here is to implement the fourth-order Runge-Kutta method, which can be used to find numerical solutions for a system of first-order ordinary differential equations.

A method to solve a coupled spring system with three springs and two masses is widely examined as a classical practical exercise.

This system is fixed at both ends: two parameters m₁ and m₂ represent masses; k₁, k₂, and k₃ are spring constants that define how stiff the spring are; and b₁, b₂ and b₃ are damping coefficients that characterize how quickly the springs' motion will stop.

Based on these equations of motion we wrote our version in terms of two coupled second-order differential equations.

The equations of motion for this system can be written in terms of two coupled second-order differential equations:

m₁ d²x₁ / dt² = -(k₁ + k₂) x₁ + k₂x₂ - (b₁ + b₂) dx₁/dt + b₂ dx₂/dt
m₂ d²x₂ / dt² = -(k₂ + k₃) x₂ + k₂x₁ - (b₂ + b₃) dx₂/dt + b₂ dx₁/dt

where x₁ and x₂ are the displacements of m₁ and m₂ respectively. There are no closed-form solutions to this set of coupled differential equations. In order to solve these equations numerically by using the Runge-Kutta method, it needs first to be converted into a series of first-order differential equations. This can be easily done by introducing the velocity variables v₁ = dx₁/dt and v₂ = dx₂/dt:

dx₁ = v₁
dx₂ = v₂
dv₁/dt = -1/m₁ (k₁ + k₂) x₁ + k₂/m₁ x₂ - 1/m₁ (b₁ + b₂) v₁ + b₂/m₁ v₂
dv₂/dt = -1/m₂ (k₂ + k₃) x₂ + k₂/m₂ x₁ - 1/m₂ (b₂ + b₃) v₂ + b₂/m₂ v₁

The initial conditions are described using the following set of parameters:

m₁ = m₂ = 0.2
k₁ = k₃ = 10
k₂ = 1
b₁ = b₂ = b₃ = 0.01
x₁(0) = 1, x₂(0) = 0, v₁(0) = v₂(0) = 0

These coupled first-order differential equations can now be solved by using the Runge-Kutta method as referenced before. The next tab shows the testing and implementation of this method.

OUR SOLUTIONS

SOLUTIONS
Differential Equations

We apply differential equations to solve case problems depending on specific requirements. We use differential equations numerical methods available such as:

Euler Method the simplest class of methods for the numerical integration of ordinary differential equations. Introduced under Euler's Method before.
Fourth-Order Runga-Kutta to achieve great accuracy with less computational effort. Introduced under Fourth-Order Runga-Kutta before.
Second-Order Differential Equationstraditionally associated with a class of methods for the numerical integration of ordinary differential equations. Introduced under Second-Order Differential Equations before

We include our real-world solution to a Hard Rock Reinforcement (Rock Mechanics) problem request.

REQUEST
An underground mining operation is challenged with serious roof collapse in their Cut-and-Fill stope mining method. Sorrounding wall rock cannot support loads over large stoping heights. Even after the stope is backfilled the work space left above still presents some risk due to falling rock.

Our task was to find a formal long term solution regarding hard rock reinforcement considering the current regional geology and stope permanence.

SOLUTION
A number of numerical techniques have been applied to problems in rock mechanics including finite element, finite difference and boundary integral methods. Difficulties arise in mumerical modeling due to complex and non-homogeneous conditions of rock mass.

This being the case, the mathematical representation via numerical modeling is a challenge of major proportions. Because of the nature of the problem, setting up and processing in terms of coupled second-order differential equations MultiRunge-Kutta4 Method (partially illustrated as sample model of execution under MULTIRUNGE-KUTTA4 METHOD and IMPLEMENTATION) we thought represented a good numerical technique to solve the problem.

After adequately formulating and modeling our computing techniques obtained successful results. Interpreting the results and formulating the correct course of action still is an on going project. This method as defined, requires a complicated set up and at this time may not represent the best solution.

TESTING

Testing the MultiRunge-Kutta Method

In order to test the first-order differential equations as defined above, a new method MultiRungeKutta4() has been added and executed:

►

           static void TestRungeKutta4();
              {
                 double dt = 0.02;
                 double t0 = 0.0;
                 VectorR x0 = new VectorR(new double[] { x10, x20, v10, v20 });
                 VectorR x = x0;
                 for (int i = 0; i < 21; i++)
                 {
                     double t = 0.1 * i;
                     x = ODE.MultiRungeKutta4(f1, t0, x, dt, t);
                     double tz = t;
                     tz = double.Parse(tz.ToString("#0.##"));
                     listBox1.Items.Add(" " + tz );
                     double b0 = x[0];
                     b0 = double.Parse(b0.ToString("#0.#####"));
                     listBox2.Items.Add(" " + b0);
                     double b1 = x[1];
                     b1 = double.Parse(b1.ToString("#0.#####"));
                     listBox3.Items.Add(" " + b1);
                     double b2 = x[2];
                     b2 = double.Parse(b2.ToString("#0.#####"));
                     listBox4.Items.Add(" " + b2);
                     double b3 = x[3];
                     b3 = double.Parse(b3.ToString("#0.#####"));
                     listBox5.Items.Add(" " + b3);
                     t0 = t;
                 }
                 this.Text = "Runge-Kutta Method for Systems / KeystoneMiningPost";
              }

Results are shown below.

DISCUSSION
Discussion of Runge-Kutta for Systems

The methods discussed in previous tabs (refer to menu) "ODE - Euler Method" and "ODE - 2nd-Order Runge-Kutta" and "ODE - 4th-Order Runge-Kutta" apply to only a single first-order ordinary differential equation as described in these tabs.

However, most problems in engineering governed by differential equations are either high-order equations or coupled differential equation systems.

A high-order differential equation can always be transformed into a coupled first-order system of equations. The trick is to expand higher-order derivatives into a series of first-order equations.

A very common example described in technical literature applies to model a spring-mass system with damping, which describes the use and calculation of second-order differential equations. We followed very closely these techniques for our own implementation.

m d²x / d t² = -kx - b dx/dt

where k is the spring constant and b is the damping coefficient. Since the velocity:

v = dx/dt

the equation of motion for a spring-mass system can be rewritten in terms of two first-order differential equations:

dv/dt = -k/m x - b/m v
dx/dt = v

In the above equation, the derivative of v is a function of v and x, and the derivative of x is a function of v. Since the solution of v as a function of time depends on x and the solution of x as a function of time depends on v, the two equations are coupled and must be solved simultaneously.

Most of the differential equations in engineering are higher-order equations. This means that they must be expanded into a series of first-order differential equations before they can be solved using numerical methods.

The next tab shows extending the fourth-order Runge-Kutta method discussed previously to a system of ordinary differential equations.

Other Applications...

Object-Oriented Implementation

Graphics and Animation

Sample Applications

Ore Extraction Optimization

Vectors and Matrices

Complex Numbers and Functions

Ordinary Differential Equations - Euler Method

Ordinary Differential Equations 2nd-Order Runge-Kutta

Ordinary Differential Equations 4th-Order Runge-Kutta

Higher Order Differential Equations