A number of problems in mechanics and other fields can be addressed by solving computer models of systems of equations that represent discretized differential or integral equations. For example, CFD applications solve Navier-Stokes or Euler equations of fluid mechanics, discretized according to finite-difference or finite-element methods (ref [1]). Solutions to these equations are most often described by fields of velocity vectors, pressure, and sometimes temperature, organized into grids of two or three spatial dimensions. Automobile designers use CFD to simulate the cabin acoustics and ventilation inside a passenger compartment under various operating conditions. Food industry manufacturers use CFD calculations to minimize the friction and heat dissipation of factory equipment. Aerospace designers use CFD to calculate lift and drag of airfoils and rockets under operating conditions that are unattainable in wind tunnels.
In a similar way, problems in structural mechanics can be solved by using finite element methods to discretize equations of elasticity and heat transfer, and a variety of other equations for problems in electromagentic scattering, radiative transfer, and other subjects (ref [2]). Most problems use spatial grids to represent the discretized equations, and typical solution algorithms iteratively refine values on the grid until the values are consistent with the governing equations. In solving these types of discretized equations, which are sometimes known as field equations, it is usually necessary to tailor the structure of the spatial grid in response to properties of the solution. For example, a CFD calculation will usually need to refine a grid by adding points around a shock wave or discontinuity in order to fulfill the requirements for consistency between the solution and the governing equations. A calculation that models the propagation of a fracture in a solid material will refine a grid to capture the structure of the fracture.
References
[1] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Springer, New York, 1991.
[2] T. J. R. Hughes, The Finite Element Method, Prentics-Hall,
Englewood Cliffs, New Jersy, 1987.
