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# Reduced integration in finite element analysis

Most finite element (FE) codes find a solution by calculating the element stiffness matrix and then inverting it to solve for the displacements in the element. For complicated finite element problems, using high order elements, it becomes necessary to use numerical integration to calculate the stiffness matrix.

Simpson's rule is a well-known method for numerical integration and can be adapted to two-dimensional integration, however Simpson's rule uses fixed intervals. There is another method called Gaussian Quadrature - this method is just as accurate but requires less function evaluations because it is not restricted to using fixed intervals. Therefore this method is used in finite element calculations.

For the function to be integrated, a number of points are calculated and their positions are optimised, known as Gaussian co-ordinates. For each of these points, the function is multiplied by an optimised weight function. Then these are added together to calculate the integral.

Reduced integration uses a lesser number of Gaussian co-ordinates when solving the integral. Clearly, the more Gaussian co-ordinates you have for each element, the more accurate your answer will be, but this has to be weighed up against the cost of computation time.

Using reduced integration will basically mean it will take less time to run the analysis but it could have a significant effect on the accuracy of the element for a given problem. Displacement-based FE formulations always over-estimate the stiffness matrix and the use of fewer integration points should produce a less stiff element. Therefore, in some cases, particularly non-linear problems such as plasticity, creep or incompressible materials, it is actually advisable to use reduced integration instead of full integration. The slight loss of accuracy is counteracted by the improvement in approximation to real-life behaviour.

Care should be taken when using reduced integration as in certain applications instability may occur due to the stiffness matrix being zero; this is otherwise known as the hour glass mode.