# Finite Element Analysis

The laws of physics are typically described by partial differential equations (PDEs), where a dependent variable such as temperature, velocity or electric potential varies with respect to an independent variable, such as position or time. Analytical solutions to these equations are often only possible for the simplest geometries which, though mathematically rigorous, rarely reflect the reality of a practical design. Modern computers provide a means of approximating solutions to these physical situations by employing numerical techniques, with the most common being Finite Element Analysis (FEA). A geometry of arbitrary complexity is spatially discretised by creating a mesh, containing nodes at which the dependent variables are solved. Basis functions are selected to represent a spatial variation in these variables, each scaled by piecewise weightings for a solution consistent with the underlying PDE. Boundary conditions are applied which represent the value of dependent variables at nodes or boundaries, for which a unique solution can then be derived.

The solution is an approximation and convergence will therefore depend on meeting a predefined criterion. A common approach is to define a maximum relative error in a dependent variable between solver iterations, of less than 0.1% for example. By using a finer mesh or increasing the order of basis functions the rate of convergence can be increased at the cost of computational time. In certain cases, it is necessary to refine the mesh where a steep gradient in the dependent variable is expected, for example when solving fluid velocity close to no-slip boundaries. A mesh refinement study can be conducted to gauge this effect in a similar manner to the relative error.

The aim of FEA software is to reduce the need for prototyping and experimentation in the design or optimisation of a device. COMSOL Multiphysics is the principal FEA modelling package used by Xi, for which Xi are certified consultants and recognised internationally. COMSOL provides the capability to solve for stationary, frequency/time-dependent and modal problems across multiple physical domains including solid mechanics, acoustics and electromagnetics. By establishing an FEA model which reflects measured data, it is possible to develop a deeper understanding of a design or device, driving further optimisation and innovation.

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