On Mining and Numerical Solutions

Author: George Pacheco


The main advantage of applying numerical analysis in mining computing is the use of mathematics rich set of algorithm libraries. These libraries implement a complete collection of mathematical, statistical and numerical algorithms which have evolved steadily for several decades. Each subroutine and algorithm in these libraries has undergone rigorous testing and implemented in real-world applications.

Algorithms for well-known math functions have been developed in numerous ways. One single premise is implemented under different configurations to produce the same results. It is precisely this flexibility that allows excellent choices when selecting appropriate software in mining projects.

When we have a number of data points obtained by sampling and experimentation, it is possible to construct a function that closely fits those data points in order to estimate the size of a mineral field available for extraction and calculate the estimated amount of mineral contained within the field. Numerical interpolation algorithms provide a way to maximize the knowledge of known ore occurrences and the method of their formation and determine potential areas where the particular class of ore deposit being sought may exist. There are various techniques for the solution of interpolation applications. Algorithms using Cubic Spline and Newton Divided Difference theory are often used.

We are cognizant that science and engineering concerns itself with the manipulation of vector and matrices. Why should we in mining be concerned with linear algebra? The concise notation introduced in linear algebra for vector and matrix operations can be directly adapted to mineral object-oriented programming. Linear algebra algorithms offer powerful operations defined for vectors and matrices. These basic techniques have been applied to a wide variety of fields in mineral engineering. Very often solutions consist of large matrices and can be used to describe linear equations and matrices that can be added, multiplied , transformed and decomposed in many ways. A very far-reaching and extremely useful tool.

Quite often optimization algorithms are used to find values of variables that yield a minimum or maximum function value. A variety of optimization methods exist that can be applied to real-world engineering problems. These type of algorithms are iterative in nature that require starting values for the variables. There are various techniques for the solution of optimization problems. In mining, this optimization technique is applied to mineral extraction where the objective function consists of minimizing the total cost of mining based on constraint parameters. Restrictive parameters like grade of ore, transportation costs, manpower and other factors are applied. Several methods are available where the Simplex Method for multi-variable functions is widely used.

Complex numbers, complex functions and complex analysis in general are part of an important branch of mathematics. They find wide application in solving real scientific and engineering problems. Complex numbers can be added, subtracted, multiplied and divided just like real numbers. In mining the implementation of trigonometric functions (trigonometric, hyperbole, inverse) are often utilized to reflect change under certain conditions. In specific situations (i.e. laborers needed per week per site) where the Sine Function Algorithm present a solid interpretation and is often used.

When dealing with measurement data inaccuracies obtained from drilling or sampling, we know it will contain significant variations due to measurement errors. The purpose of curve fitting solutions is to find a smooth curve that on average fits the data points. There is a distinction between interpolation (see Interpolation above) and curve fitting. Curve fitting is applied to data that contain gaps, usually because of measurement errors and tries to find the best fit to a set of given data. The curve does not necessarily pass through all the given data points. The Straight Line Fit Method and a Polynomial Curve Fitting Method for distinct order-polynomials are widespread utilized algorithms.

Many scientific and engineering phenomena are characterized by nonlinear behavior and solutions of nonlinear applications and is a fundamental issue in engineering analysis.The simplest case to find the single root of a single nonlinear functions is widely used in underground mining operations where situations call for special requirements in construction (i.e platform at a slope, or ventilation equipment , or similar patterns). The Newton-Raphson Method> is one of many used in industry. Fixed Point and Birge-Vieta methods are also popular.

Refreshing our high school math, we may recall that numerical differentiation deals with the calculation of derivatives of a smooth function . In mining, numerical differentiation occasionally is used to calculate displacements due to exploratory drilling or similar activities where terrain displacements may show different stress concentrations considered under uniform stress. To find the stresses and hence the stress concentration factor, it is necessary to find the derivatives of those displacements. The Forward Difference Method is one of many algorithms used to accomplish this task. Other algorithms like Backward Difference Method, Central Difference Method, Extended Central Difference Method, Richardson Extrapolation, Derivatives by Interpolation are also extensively used.

When it comes to solve problems that involve given starting conditions, such as, volume, time, space and other parameters, differential equation techniques.are used. Several numerical methods exist to solve ordinary and high-order differential 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.

The use of differential equations in mineral engineering is extensive. The scope of applications is as diverse as evaluating tasks quantifying the grade and tonnage of a mineral occurrence to everyday complexity encountered in open pit mining or underground shaft sinking, block caving, cave mining , cut and fill, or similar extraction operations. Many commercial mineral algorithms are available based either on Euler Method or 2nd –Order Runge-Kutta Method or 4th –Order Runga-Kutta Method.

Problems originating in mining engineering often require the solutions of differential equations in which the data to be satisfied are located at two different values of an independent variable. These are called the boundary conditions and these algorithms approximate the differential equation by finite differences at evenly spaced mesh points. Now and then this technique is used in tunneling where the finite difference method is particularly suitable for linear equations. Commercial algorithms exist such as, Shooting Method, Finite Difference Method, Finite Difference for Nonlinear Method, Finite Difference for Higher-Order Method and others.

To put it briefly, numerical solutions are included at every stage of a mining operation and the descriptions of algorithm utilization above represent a very limited view of endless applications. The use of numerical analysis makes mining planning a much more organized and efficient form of mineral extraction. Math algorithms in mining provide the industry with extraordinary tools to assist time-consuming tasks be reduced to manageable units to obtain solutions which otherwise would be very difficult to achieve.

Practical samples and further interesting discussions on mineral engineering can be found at http://www.keystoneminingpost.com


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