Algorithms for well-known mathematical functions have been developed in numerous ways. One single postulate is implemented under different configurations to accomplish the same results. It is precisely this flexibility that allows excellent choices when selecting appropriate software in mining projects. The application of numerical analysis in mining is essential where the selection of the best, most accurate subroutines will assist to process a solution faster.
Consider for example exploratory drilling in geology. 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. There are various techniques for the solution of interpolation applications where algorithms based on 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? Linear algebra algorithms offer powerful operations defined for vectors and matrices, where the concise notation for vector and matrix operations can be directly adapted to mineral object-oriented programming. Very often solutions consist of large matrices that can be used to describe linear equations where matrices 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. 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. 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 fluctuations (terrain, faults) under certain conditions. Sine and Cosine Function based algorithms are 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. Curve fitting is applied to data that contain gaps and tries to find the best fit to a set of given data where 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 underground 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.
We may recall that numerical differentiation deals with the calculation of derivatives of a smooth function (by bringing back our high school math). In mining, numerical differentiation occasionally is used to calculate displacements sometimes due to exploratory drilling or similar activities where terrain displacements (or any other discrepancy) may show different stress concentrations considered under uniform stress. To find the stresses and hence the stress concentration factors, it is necessary to find the derivatives of those displacements. Many algorithms exist: The Forward Difference Method, Backward Difference Method, Richardson Extrapolation, Derivatives by Interpolation.
When it comes to solving 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. 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, cut and fill, or similar extraction operations methodologies. Many commercial mineral algorithms are available based either on Euler Method or 2nd-Order Runge-Kutta Method or 4th-Order Runge-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. On occasion this technique is used in tunneling where the finite difference method is particularly suitable for linear equations. Commercial algorithms based on uch as, Shooting Method, Finite Difference Method, Finite Difference for Nonlinear Method, Finite Difference for Higher-Order Method and others.
Based on the few samples of algorithm utilization we can surmise that numerical solutions are attended to at every stage of a mining operation. The use of numerical analysis makes mining planning a much more organized and efficient form of mineral extraction. Mathematical 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.
